Not so much for the starting job. Mayfield had already locked that up, having quarterbacked the Sooners to the College Football Playoff during the 2015 season, weeks before Murray arrived in Norman, Oklahoma, after transferring from Texas A&M.
But that didn't stop them from finding other ways to compete, even in the smallest of ways.
"They're out there in a warm-up drill, seeing who could throw the best spiral," Oklahoma coach Lincoln Riley recalled. "It was just constantly about that."
Was it ever, according to Austin Kendall, Tanner Schafer, Reece Clark and Connor McGinnis -- the four other Oklahoma quarterbacks who shared a meeting room with Mayfield and Murray during those 2016 and 2017 seasons.
"We always turned every single drill into some sort of competition -- just because they wanted to beat each other, which was obviously quite fun," McGinnis said. "But oh my gosh, it was nonstop."
This Sunday, the two former teammates will be out to beat each other once again, as Murray and the Cardinals travel to face Mayfield's Browns in Cleveland (4:05 p.m. ET, Fox). In their first NFL meeting two seasons ago, Murray got the best of Mayfield. Now, Mayfield will look to return the favor by handing the undefeated Cardinals their first loss.
"Obviously he was the guy, but I gave him s--- all the time," Murray said. "It was competitive, but at the same time, we had the best quarterback room in the country -- we knew it.
"The confidence, the swagger about that room was unmatched. ... and it was fun."
Unmatched, indeed. Mayfield and Murray became the first quarterbacks from the same school to not only win the Heisman Trophy in back-to-back seasons but go No. 1 overall in consecutive NFL drafts.
"Really special," said former Oklahoma coach Bob Stoops, who handed the reins over to Riley in 2017, one year before Mayfield would do the same with Murray. "Just two incredibly talented, driven individuals that really were great examples for the team on how to be teammates, how to work and how to prepare."
Yet despite that drive to be the best, Mayfield and Murray never allowed a potentially combustible situation at Oklahoma to spill into controversy, even after encountering a pitfall or two.
Mayfield embraced the competition, as opposed to feeling threatened by Murray's unique ability. And even though he might have been a starter almost anywhere else in college football, Murray never became disgruntled or a distraction for Mayfield or the Sooners, even after Mayfield retroactively was given an extra season of eligibility, which left Murray on the sidelines.
"It was a healthy relationship for us," Mayfield said. "All egos were put to the side. ... And we were able to push each other and absolutely get the best out of each other."
They channeled those competitive streaks to help the other ultimately reach greatness -- while having fun along the way.
"For having two players of that caliber in the same meeting room, it was about as positive as you could imagine, because there's obviously a million different things that could've gone wrong where it wouldn't have been good for one or the other or both," Riley said prior to last weekend's Red River Showdown, when he benched former five-star recruit Spencer Rattler, in favor of another, Caleb Williams, who propelled the Sooners to an improbable comeback victory over Texas.
"They didn't take s--- from each other, but it was a healthy competition and a healthy respect," Riley said of Mayfield and Murray. "And they were both smart enough to figure out that they could learn something from the other.
"They certainly would've been good players anyway. But they pushed each other to become even better."
WHEN MAYFIELD AND Murray were both at Oklahoma, practice seemed almost as exciting as the games.
"Baker would be just yelling the entire day, swatting the defensive coaches on the butts" after big plays, Stoops said. "Unbelievable the energy he brought, made it fun for everybody."
Murray was as unbelievable operating the scout-team offense, virtually untouchable whenever he would scramble out of the pocket. Stoops said he and his assistants would rewatch those practice highlights, simply for entertainment value.
"It was just like, holy cow," Stoops said.
When Mayfield and Murray weren't tormenting Oklahoma's defenders, they were going at one another, competing to see who could bounce the most throws off the goalpost's crossbar or land more passes into a trash can. The quarterbacks would always split teams by even and odd jersey numbers, so Mayfield (wearing No. 6) and Murray (No. 1) could be on opposing sides; losers usually had to carry the winners' shoulder pads inside after practice.
"Here we are in the middle of special teams, a chance for those guys to relax, chill a little bit, and instead they're going all out, trying to win these quarterback games," said Ohio University assistant coach Tyler Tettleton, a former graduate assistant who worked with the quarterbacks at Oklahoma. "We looked more forward to that over anything else in practice, just because of how fun and competitive those two guys made it."
That carried over into the meeting room, where the quarterbacks perpetually squared off in a celebrity guessing game they called "passcode."
"Their level of competitiveness was not like anything I've ever been around. And I think that's what made us so good those years. Everybody was feeding off those two guys. Not just the quarterback room, but the whole team." Tanner Schafer
Oklahoma QB 2017-present
"Say me and Kyler were on a team, me and Baker would come up with a celebrity, and we each took turns saying one word to our partner to try to get them to guess it -- it just bounced back and forth until somebody guessed the celebrity," Clark said. "And we would play this for weeks on end, because they're like, 'OK we're playing to 10. No, no, no, we're playing first one to 20. No, no, first one to 30.' It just kept going on and on, just because they wanted to win even in something like that."
When Mayfield and Murray weren't playing passcode, the two would argue about almost anything, from who would win the Masters that year, to who was better in the video game Rocket League, to whose Texas high school alma mater had the superior football program.
"Their level of competitiveness was not like anything I've ever been around," said Schafer, who is still on the Oklahoma roster. "And I think that's what made us so good those years. Everybody was feeding off those two guys. Not just the quarterback room, but the whole team."
It was a mutual bravado that also led to plenty of trash talk.
"He might show it differently," Mayfield said of Murray, "but behind closed doors, he is quite the trash talker. Don't let him fool you."
Mayfield teased Murray for carrying around a keychain with a tiny bottle of sriracha sauce attached to it, which he would pop open to spice up his meals. Murray never passed up an opportunity to take a shot at Mayfield's speed -- or, in his opinion, lack thereof.
"We'd be watching film and Baker would take off on a scramble," Clark said, "and Kyler would kind of smirk, like 'Oh, I would've taken that to the house.'"
Yet while they enjoyed goofing around, Mayfield and Murray were all business when it mattered.
"They knew every answer to every question," Clark said. "Every blitz that was coming, every protection, every line shift, like the back of their hand."
And through that, Riley believes, they also learned from one another.
"The preparation, Baker is pretty meticulous with that, and I think Kyler took a lot from him," Riley said. "But I think Kyler had the ability in his mind to keep things very simple. And I think at times with Baker -- sometimes with quarterbacks, you cannot know too much, but you can be trying to process too much at times -- the way Kyler was able to break things down and keep them very simple and keep the main thing the main thing, I think that's something that Baker took note of.
"And all of that was fun to watch unfold."
IN 2015, AS Mayfield guided Oklahoma to the playoff, Murray, then at Texas A&M, was looking to transfer.
Because Mayfield had only one season of eligibility remaining at the time, the Sooners were looking to take a transfer quarterback. So Stoops had Riley, then in his first year as Oklahoma's offensive coordinator, study tape of three prospective quarterbacks -- Will Grier, Kyle Allen, who have both since started games in the NFL, and Murray -- to determine which to pursue.
"Lincoln comes back and says, 'I like all three. All three can help us and can win for us,'" Stoops recalled. "But this guy Kyler Murray, he can win a Heisman Trophy.'"
Sure enough, Murray would do just that in his lone season starting for the Sooners.
"Sometimes you see things over and over," Riley said, "and then all of a sudden you see something that's just totally different and it just hits you in a different way. That's how his tape hit for me."
By that point, Mayfield was entrenched as the starter. But Murray's natural talent never allowed Mayfield to feel complacent.
"Baker deep down knew that he had to continue to play at a high level," Riley said, "just because he had another good player in the room."
But Mayfield welcomed the challenge.
"He had the self-confidence to say, throw whoever in front of me, I'm going to take him down," Clark said. "And Kyler was so self-confident, he knew, even if he only had a year, he was going to make it all happen."
When Murray initially picked Oklahoma, though, he assumed he'd have two years to play. But the summer after the 2015 season, the Big 12 Conference voted to give Mayfield a year of eligibility back, since he walked-on to Oklahoma when he transferred from Texas Tech. The Sooners privately worried Murray might leave as a result.
"But he didn't flinch," Stoops said.
Murray redshirted during the 2016 season. Then in 2017, he was the backup, as Mayfield won the Heisman.
"Bake's a great player. He gets a lot of hate and all that, but I watched firsthand each and every day, how hard he worked, make throws that a lot of people can't make," Murray said. "I'm glad that I got to learn from him throughout those years."
Murray finally got his chance, ironically, in Mayfield's final college home game. Mayfield had been suspended from starting against West Virginia for grabbing his crotch and yelling, "F--- you!" to the Kansas sidelines the weekend before, after the Jayhawks snubbed Mayfield's handshake attempt at midfield following the pregame coin toss. That put Murray behind center on the opening series against the Mountaineers.
Before the first snap, Stoops told those in his stadium suite to buckle up. And sure enough, on the very first play, Murray took off for a 66-yard run before being dragged down at the West Virginia 4-yard line.
"And the person most excited on our sideline is Baker," Riley said. "That was just them, how they were."
The next day, though, as they watched the film, Mayfield couldn't resist lobbing a "not fast enough" jab back at Murray for being caught from behind.
"The respect those guys had for each other went a long way," said Austin Kendall, now Louisiana Tech's starting quarterback, who noted how much he learned from Mayfield and Murray about leadership and preparation. "And I respected the hell out of those guys for it."
WEEKS LATER, IN the aftermath of the season-ending-defeat to Georgia in the Rose Bowl, Mayfield stood a few feet away from Murray's locker. As soon as he finished answering questions about losing the heartbreaking overtime thriller, he beelined to Murray and gave him a hug.
"They're in great hands," Mayfield had said moments before of the Sooners. "Kyler is the best athlete in the country. ... They're going to be just fine."
Just as Mayfield had predicted, Oklahoma was more than fine. Murray went on to break Mayfield's FBS single-season passing efficiency record while winning the Heisman and leading Oklahoma back to the playoff.
"You could just tell he had it -- the way he elevated his teammates, the way he played," Mayfield said. "He might show it in a different way, but he's extremely competitive. That's another reason why he is where he is."
Their shared success from college has since carried into the pros.
Last season, Mayfield led the Browns to their first playoff victory in 26 years. And behind Murray's spectacular start, Arizona is the NFL's last unbeaten team this season.
"They were a lot of fun to have in the room. ... And I think they ended up getting the best out of each other," Riley said. "They're the ones that deserve the credit because they handled it in a way that a lot of people wouldn't have."
ESPN Arizona Cardinals reporter Josh Weinfuss contributed.