A source told the newspaper Brown had brought up the idea of resigning to university officials, and ESPN.com said its sources reported Brown could announce his departure by the end of the week.
"The talks were very friendly and the conclusion was Mack would step down in the next couple of days," ESPN's source said.
Brown has led the Longhorns for 16 seasons and is under contract through 2020 and Athletic Director Steve Patterson said in a written statement Tuesday that Brown was still the team's head coach. "We continue to discuss the future of Texas football. Mack Brown has not resigned. And, no decisions have been made."
The Statesman-American said it heard from eight different sources that no one at Texas was pressing Brown to make a decision or had set a deadline for him to decide if he had had enough. Sources said Brown had not said anything to his coaching staff.
Brown was scheduled to take part in a news conference Thursday as part of the run-up to the Alamo Bowl. The Longhorns will face Oregon in the Dec. 30 game.