The deal is for six years, a source told ESPN's Adrian Wojnarowski.
Georgia athletic director Greg McGarity described Crean as "one of the most successful coaches in college basketball over the past two decades.''
"His teams have consistently been participants in postseason play, and his players have been extremely successful in the classroom," McGarity said in a statement. "He's going to be a great fit for the University of Georgia. I'm extremely excited to have him leading Georgia Basketball into the future and to welcome his family into the Bulldog Nation."
Crean, who will turn 52 on March 25, spent nine seasons at Indiana before being let go after an 18-16 campaign in 2016-17. He has spent this season working as an analyst for ESPN.
"I am honored and humbled to join the University of Georgia family," Crean said. "I am sincerely grateful to President Morehead and Greg McGarity for an incredible opportunity. Make no mistake, this is a basketball program inside of a great university that can compete for championships doing it the right way."
Georgia has scheduled a Friday news conference to introduce Crean. It will be at 3 p.m. ET.
Crean's hiring comes one day after former Ohio State coach Thad Matta withdrew from the Georgia coach search after becoming the first known candidate to interview for the job.
Fox was fired Saturday after earning two NCAA tournament bids in nine seasons.
During Crean's time in Bloomington, the Hoosiers went to four NCAA tournaments, reaching the Sweet 16 on three occasions. Crean also spent nine seasons at Marquette, reaching the Final Four in 2003. He made five NCAA tournament appearances with the Golden Eagles.
Crean also was an assistant coach under Ralph Willard for one season at Pittsburgh in 1994-95.
Georgia (18-15) lost to Kentucky in the quarterfinals of the SEC tournament, ending the Bulldogs' hopes of landing a spot in the NCAA tournament.
Information from ESPN's Jeff Borzello and The Associated Press was used in this report.