Davis was expected to be introduced as the new coach at an afternoon news conference, said the high-ranking university source, who spoke on condition of anonymity.
"I got the job," Davis told ESPN.com.
The word that the university had chosen Davis came two days after Indiana fired Bob Knight for misconduct and a day after players issued an ultimatum to athletic director Clarence Doninger that Davis or John Treloar, another Knight assistant, be hired as interim coach or members of the team would defect en masse.
The university source told The Associated Press that Treloar also will remain with the program. The source said he did not know the status of Pat Knight, the son of the Hall of Fame coach and also an assistant.
The university source said the hiring of Davis was expected to have a "very positive" impact on the storied basketball program, which had appeared in disarray.
Several players had threatened to leave the program. Junior guard Dane Fife said Monday he had decided to transfer, but the university source said Fife's departure was not yet certain.
Highly recruited freshman swingman A.J. Moye of Atlanta, Ga., also had indicated Monday he likely would leave Bloomington.
Athletic director Clarence Doninger had said Monday he had asked Davis and Treloar to remain. He also said he had received calls from "a number of people" interested in the interim post. He would not identify any of the others.
Despite Knight's rigid discipline, uncompromising demand for perfection and infamous temper which got him fired Sunday he was the main reason players have come to Indiana to play basketball over the past 29 years.
Doninger understood why the players were upset.
"You've got some quality people and they are stunned by all of this" he said. "There's no question they came to Indiana University to play for coach Bob Knight. We'd like to think they came to Indiana University because of Indiana University, too, but I know the facts of life here, and they came to play for coach Knight."
Knight was fired by university president Myles Brand for violating a "zero tolerance" behavior policy imposed in May.
It's not known what Knight's next move will be, but Indiana Pacers coach Isiah Thomas who led Knight's 1981 team to a national title said he would welcome Knight as an assistant.
"I would love for him to sit on the bench with me and more or less mentor me," Thomas told Detroit radio station WDFN. "I don't think there's a basketball player in the world who wouldn't crave his insight. That's what, hopefully, he'll be able to give me."
The Mavericks' owner, Indiana graduate Mark Cuban, told the Fort Worth Star-Telegram he'd be happy to have Knight work for him.
"As a huge fan, Bobby has a standing offer to join the Mavs as a consultant," Cuban was quoted as saying in Tuesday's editions of the newspaper. "I would offer him a job in a heartbeat."
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