Kelly, 49, has served as the head coach at Oregon since the 2009 season and was the team's offensive coordinator for the two seasons prior. He had coaching experience at New Hampshire, Johns Hopkins and Columbia in the 17 seasons prior.
On Monday, the Browns fired general manager Tom Heckert and head coach Pat Shurmur after the team finished with a 5-11 record. The Browns' offense ranked No. 24 in the NFL in points scored and No. 25 in yardage gained. The team has just one winning season in the last 10 years, but has not made the playoffs since 2002.
Oregon defeated Kansas State in the Fiesta Bowl on Thursday night, 35-17.