Tressel, who resigned in May after an NCAA scandal that enveloped the Ohio State football program, was hired by former Colts coach Jim Caldwell in September to be a game-day consultant in a role that allowed him to work from the team's coaches booth and advise Caldwell about replay reviews.
Irsay wrote on his Twitter account Saturday that he expects to decide on a new coach next week.
Tressel didn't begin his duties with the Colts until the seventh game of the season.
NFL commissioner Roger Goodell has said he would have suspended Tressel if the Colts didn't make him sit out games to start the 2011 season. Because of the NCAA violations involving improper benefits, Tressel would have served a five-game suspension to start Ohio State's 2011 season had he remained coach of the Buckeyes.
On Friday morning, Irsay flew to Sarasota, Fla., where Tressel owns a home, according to the Indianapolis Star, and according to WBNS-TV in Columbus, Ohio, a second interview was conducted.
Irsay has gutted the Colts' front office and coaching staff after their 2-14 season, and Caldwell was fired Tuesday, two days before eight assistant coaches were let go.
Irsay's tweet read: "The (hash)1 pick debate will rage on,what a great year to have it..the HC search is wide ranging n thorough,decision by mid 2 late next week."
The Colts asked for permission to interview Bengals offensive coordinator Jay Gruden, but Gruden declined the interview, a league source told ESPN NFL Insider Adam Schefter.
The Colts played the season without Peyton Manning, who had his third neck surgery in September.
The day after they locked up the No. 1 overall draft pick, Irsay fired team vice chairman Bill Polian and his son, general manager Chris.
Information from The Associated Press was used in this report.