UT's board of regents will discuss the matter in a specially called Sunday meeting that will include consultation with attorneys.
In separate statements released Friday night, Dodds and Applewhite called the incident a one-time occurrence. Dodds said it happened during activities related to the 2009 Fiesta Bowl, when Applewhite was UT's assistant head coach and running backs coach.
Dodds said he learned of the incident later that month, and that Applewhite admitted his "inappropriate conduct." Applewhite "fully accepted his discipline, including counseling," Dodds said.
"Several years ago, I made a regretful decision resulting in behavior that was totally inappropriate," Applewhite said in his statement. "It was a one-time occurrence and was a personal matter. Shortly after it occurred, I discussed the situation with [Texas athletic director] DeLoss Dodds. I was upfront and took full responsibility for my actions. This is and was resolved four years ago with the university.