Quarterback Joe Burrow, wide receiver Tee Higgins and linebacker Logan Wilson are up for extensions this year, while two-time Pro Bowl wide receiver Ja'Marr Chase is eligible for a new deal next year.
At its annual media luncheon Monday, Brown said keeping all of the team's top players could be a challenge given the salary cap constraints.
"We have some good players that need to be fed," Brown said. "That's a challenge. It's mathematical. You get a bag of corn and you have 10 hogs. Well, you're going to put that out to them. The bag's going to be empty. And some of them aren't going to get it."
Burrow is entering his fourth year in the NFL and is coming off his best season as a pro. In 2022, Burrow was 10th in QBR and third in completion percentage over expectation, according to ESPN Stats & Information.
Throughout the offseason, both sides have remained mum on Burrow's contract situation. Brown and team director of player personnel Duke Tobin did not divulge any specifics Monday on the contract status for the franchise quarterback.
"I have bound myself not to talk about Joe's contract," Brown said. "I don't think it's helpful for the negotiations. The other side has made the same commitment and they have not broken it."
According to the team, Burrow was present Sunday when rookies conducted their first drills of training camp. Veterans are scheduled to report for camp Tuesday, with the first practice slated for Wednesday.
Higgins has amassed 3,028 receiving yards in his first three seasons and participated in the team's voluntary workouts despite the lack of a long-term extension. In a post issued Monday on Threads, Higgins indicated he was set for camp, saying he was "ready to get back to work."
Tobin remained optimistic that the team will be able to retain Higgins and Chase as part of its offensive core in upcoming years.
"We're not out looking to replace them with other guys," Tobin said Monday. "We'd like to keep them around. We'll see if that's possible. I can't predict that, but we're definitely hopeful that we have long-term relationships with the guys."
While Tobin didn't give any details about the contract negotiations with Burrow, the team's longtime personnel chief said the quarterback "understands the big picture and we understand the big picture."
Cincinnati is trying to win its first Super Bowl ever after coming close the past two seasons. In January 2022, the Bengals lost Super Bowl LVI to the Los Angeles Rams 23-20. One year later, Cincinnati fell by the same margin in an AFC Championship Game rematch against Kansas City.
Brown, who has been the head of the franchise since his father, team founder Paul Brown, died in 1991, said he wants to be able to celebrate hoisting the Lombardi Trophy in Cincinnati.
"Zac [Taylor] is a great head coach," Brown said. "We have good players. Our quarterback is an exceptional player. We have a shot and we'll see where it goes this year, but it would be fun if we could put it together."