The agent, Jimmy Sexton, made the comments during a 45-minute call with former Texas Regent Tom Hicks and current Regent Wallace Hall. Hicks detailed the call in a Sept. 24 email that was obtained Tuesday by the AP through an open-records request.
"Sexton confirmed that UT is the only job Nick would possibly consider leaving Alabama for, and that his success there created special pressure for him," Hicks wrote.
The pressure was not further explained in the email and Sexton declined comment Tuesday. Saban was not available after practice.
Two days after the call, Hicks approached Texas coach Mack Brown about the possibility of retiring. Brown, who is under contract until 2020, said he wanted to stay.
Hicks sent the email to his brother and current Regent Steve Hicks five days after the AP first reported that the call took place.
Saban has said he didn't know anything about the meeting, and he's too old to start over someplace else. The Tide has won the last two national titles and his team is currently ranked No. 1.
Brown, who will be paid $5.4 million this year, won the 2005 national title and lost to Saban's Alabama team in the 2010 championship game. The Longhorns are 28-18 since that defeat.
Brown has said he plans to coach through his contract. But three sub-par seasons and 1-2 start this year raised intense speculation about his future.
Some of that pressure has eased after a five-game winning streak that has Texas in first place in the Big 12, but Saban is often mentioned as a potential replacement.
Saban has won four national championships, one with LSU in 2003 and three with Alabama after the 2009, 2011 and 2012 seasons. Saban earns $5.6 million per year, but Texas — the nation's wealthiest athletic program — certainly could afford him.
On Tuesday, Texas announced it had hired Steve Patterson to be the school's new athletic director. He replaces DeLoss Dodds, a staunch Brown ally who is leaving after 32 years.
It is still unclear who initiated the contact between Sexton and the Texas representatives, although Tom Hicks' email suggests it was Sexton. Hicks wrote that Steve Hicks had told him "an agent had made contact to say Saban could have interest in UT."
Tom Hicks also wrote that Wallace Hall had told him that a "very confidential friend (of Hall) had heard from Jimmy Sexton, Saban's agent, and that he was willing to have a call."
Hall told the AP in September that a person he would not identify called him, unsolicited, and proposed an introduction to Sexton. Hall, who is under impeachment investigation by state lawmakers, could be asked to detail the meeting and identify the go-between if questioned under oath next week.
Steve Hicks, who initially told the AP he had never "authorized anyone" to talk to Sexton but later acknowledged his involvement, traded several emails and texts with the brother about setting up the meeting, Tom Hicks wrote.
Tom Hicks and Hall spoke with Sexton by speakerphone at Hall's home.
"We told Sexton that Mack had leadership's support to stay and that I would go talk to him as a friend to see if Mack had any interest in retiring," Tom Hicks wrote.
"I told him it would have to be Mack's idea," he wrote.
Tom Hicks met Brown over lunch to tell Brown about Saban and "wanted him to know of Saban's interest if Mack wanted to make it his idea," Hicks wrote.
"Mack was adamant that he had no interest, and that he wanted to go out on top and leave UT in great place," Hicks wrote.
Hicks wrote that Brown was upset at Hall "for initiating the process" and called his lawyer Joe Jamail for legal advice. Jamail has threatened to sue anyone outside of the university system if they try to meddle with Brown's contract or job status.
Hicks said he assured Jamail he had gone to see Brown as a friend and not as a representative of the board of regents.
"I told Mack I was pleased to see his competitive instincts aroused and recommended he share his enthusiasm more with the fans," Hicks wrote.
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