Kelly has been at or near the top of the Browns' list of candidates since the team fired Pat Shurmur, who went 9-23 in two seasons. Cleveland owner Jimmy Haslam and CEO Joe Banner have been conducting interviews in Arizona all week, searching for the team's sixth full-time coach since 1999.
Cardinals defensive coordinator Ray Horton confirmed he interviewed with Cleveland earlier this week. The Browns have reportedly met with former Arizona coach Ken Whisenhunt, Marrone and Penn State's Bill O'Brien, who removed himself from any consideration on Thursday night and intends to stay at the school.
The Browns might have been interested in Alabama's Nick Saban, but the 61-year-old doesn't seem to have any interest in returning to the NFL after two seasons with Miami.
Kelly doesn't have any pro coaching experience, but aspects of his up-tempo offense are already being used by some NFL teams, including New England and Washington.
Kelly wouldn't say if he was leaning one way or another following the Ducks' bowl win.
"I said I'll always listen, and that's what I'll do," he said. "I know that people want to talk to me because of our players. The success of our football program has always been about our guys. It's an honor for someone to say they'd want to talk to me about maybe moving on to go coach in the National Football League. But it's because of what those guys do. I'll listen, and we'll see."
Oregon could be facing possible NCAA sanctions for the school's use of recruiting services, but Kelly indicated he isn't running from anything.
"We've cooperated fully with them," he said. "If they want to talk to us again, we'll continue to cooperate fully. I feel confident in the situation."
Oregon's players gave Kelly a Gatorade bath as the final seconds ticked off the clock in Thursday night's game, and afterward a few of the Ducks seemed resigned to their coach moving on.
"We'll have to see," quarterback Marcus Mariota said. "Whatever he decides to do, we're all behind him. He's an unbelievable coach. He's not only a football coach, but he's someone that you can look to and learn a lot of life lessons from. Whatever happens, happens. But we're all behind him.
"We'll see where it takes us."
AP Football Writer Rob Maaddi in Philadelphia contributed to this report.
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