"It looks as though it's going to happen," one source told ESPN. "It's close, but it's not done yet."
Barnes, 60, was fired by Texas athletic director Steve Patterson over the weekend despite going to 16 NCAA tournaments in 17 seasons.
On Sunday, Barnes confirmed that Patterson told him a few days ago he had to fire members of his coaching staff or be fired himself. That ultimatum was leaked to the media Thursday, publicly turning up the pressure on Barnes and his assistants. Barnes blamed the leaks on the athletic department but didn't name any individuals behind them.
"I was shocked" by the leaks, Barnes said. "I couldn't do that. That would be me saying this is about me. I've been carried by a lot of people here. We're in this together."
Barnes said his staff offered to quit, but he refused to let them.
The Longhorns did not live up to expectations this season, barely sneaking into the NCAA tournament before being eliminated by Butler in their opening game.
Barnes would replace Donnie Tyndall, who was fired after one season because of unethical conduct at his former post, Southern Mississippi.
The Associated Press contributed to this report.