The ESPN video, released Friday, was taken during a relaxed interview session at the College Football Awards show Dec. 8.
Garrett attended the show at the College Football Hall of Fame in Atlanta and spoke to an ESPN social media producer, who asked whether there was a specific team he might want to join in the NFL. His answer was "100 percent supposed to be fun," the producer said.
Garrett, a Texas A&M defensive end who hadn't officially declared for the NFL draft at that time, addressed Jerry Jones, owner of his hometown Cowboys, and Dallas coach Jason Garrett.
"All right, I'm speaking to you, Jerry," Garrett said. "Mr. Garrett, make it happen. Dak Prescott is leading our team right now. I need you to take Tony Romo, take a couple picks, give them to Cleveland so you can pick me up. Please. I would love to play in Dallas. Just make it happen."
Garrett on Saturday said he's been caught off guard by all the attention the comments have received.
"It was supposed to be pretty much a joke and not taken too seriously," said Garrett, who is in Houston on Saturday for the Lombardi Award ceremony. "It kind of got blown up."
He later added: "Take a joke, people."
The social media producer said Garrett's comments were in the spirit of an upbeat event. All players in attendance went through a series of interviews before the event.
"He was definitely having fun and was in a playful mood," the producer said. "One of the questions right before that was about his interest in dinosaurs. That's what we do. We get fun little nuggets.
"It was 100 percent supposed to be fun. He was just having fun. He knew I worked for ESPN and that I was videoing him."
In a separate video for KRIV-TV in Houston that was posted on Twitter on Thursday, Garrett said he would "definitely" like to play for the Browns.
"People might say, oh they're this, they're that, or I made a comment about cold weather and they kind of pointed toward Cleveland," Garrett said. "It doesn't matter to me. I'll play wherever they put me. It's about your mindset you go into it [with]. If you go in there with a mindset that you're going to turn things around, you can make that contagious and people start to believe in it, then you can turn it into a winning program wherever you go."
Garrett grew up in Arlington, where the Cowboys play. He is projected to be one of the top two or three picks in this draft, perhaps the first overall.
"I've already said I like the Cowboys because that's my hometown team," Garrett told ESPN on Saturday. "Everybody knows you're going to like your hometown team. But I want to go No. 1, whoever that is. If that's Cleveland, I have no problem going up there and playing with them. I'm going to love whatever team and organization that I'm a part of.
"... Cleveland, Chicago, San Francisco, whoever has No. 1 at the end of the day, that's where I'm trying to go."
The Browns have the first pick and need a quarterback, but it would be highly unlikely that a rebuilding Browns team would trade for Romo, who is 36.
It's even more unlikely that the Browns would alter their draft plans based on a video like this one -- especially given when it was taped.
Garrett on Saturday said the trade was just something he and his friends had joked about, but it was never meant to be taken seriously.
"I thought it would be funny to put that in the video, because the video wasn't supposed to be serious," Garrett told ESPN on Saturday. "(My friends said) 'Well maybe they can (trade) Romo or trade all of their picks.' I said, 'That'd be one hell of a trade.' But no, [I wasn't being serious]. I just want to go No. 1."
Garrett attended the banquet because he was one of three nominees for the Chuck Bednarik Award, given to the defensive player of the year in college football. Defensive lineman Jonathan Allen of Alabama won the award.
ESPN's Sam Khan Jr. contributed to this report.