"We've made a decision. He's our coach," said Ohio State President E. Gordon Gee.
Gee said he talked to the coach for hours about his mistake, but said he never considered firing him, 10 Investigates' Paul Aker reported.
The university has imposed a two-game suspension and a $250,000 fine, after Tressel failed to disclose that he was informed that two of his players were linked to a man who was being investigated by federal agents, 10TV News reported.
While the e-mails showed that Tressel learned about what was going on in April, he did not disclose it to the university until January, when university attorneys discovered it. Ohio State self-reported it to the NCAA on Feb. 3.
During a news conference on Tuesday, Gee quipped that he was "glad he didn't dismiss me" in response to questions about whether he considered dismissing Tressel.
The comment was widely quoted and on Thursday Gee admitted it was not a comment he would make again.
"Oh, it probably no, but I am a guy who speaks in ways that I generally have a little bit of humor to what I'm saying," Gee said. "If I had to say it again, I wouldn't."
Gee was willing to repeat his faith in Tressel and his belief the university will move forward.
He said while Tressel made a mistake, he is confident no other problems will emerge as an NCAA investigation moves forward.
"We would not be confident in what we've done if there was a smoking gun," Gee said. "Is there another shoe to drop? Absolutely not."
Although the university issued a self-imposed two-game suspension, the NCAA will make the final decision on any punishment Tressel will receive.
The NCAA has not said how long it will take to complete the investigation.
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March 9, 2011:
Ex-Ohio State Football Player Says He Told Tressel About Player Investigation
March 9, 2011: Tressel's Rules Violation Could Impact Ohio State Recruiting
March 8, 2011: Ohio State Suspends Coach Jim Tressel For 2 Games
March 7, 2011: Report: Tressel Knew About Violations Last April
December 30, 2010: NCAA Rebuts Critics Of Ruling On Ohio State Violations
December 28, 2010: Suspended Ohio State Football Players Publicly Apologize
December 23, 2010: Pryor, 5 Others, Face Sanctions For NCAA Violations
December 22, 2010: Ohio State Probing Possible NCAA Violations By Football Players