Dungy made the comment during a live look-in at Dan Patrick's radio show during its television simulcast on the Audience Network.
"I don't know what's going on there, I really don't," Dungy said during the look-in. "Is he going to play? Jim Irsay made a comment about six weeks ago, 'You know, it's inside his head now.'"
Patrick followed up by asking the former Colts coach: "Wait, it's inside Luck's head?"
"Yeah, he said that I guess when I was up there for Peyton's [Manning's] ceremony," Dungy said.
Dungy on Tuesday clarified with ProFootballTalk.com that Irsay didn't make the comments to him privately.
"I never had a conversation with Jim Irsay," Dungy told the website. "I never said I had a conversation with Jim Irsay."
The only time Irsay has publicly talked about Luck needing to overcome mental hurdles, however, was on Aug. 31, following the Colts' final preseason game, when he addressed approximately 10 to 15 reporters.
"It's a great question, because it's been said before by one of the greatest athletes and competitors who've played any sport -- the quote was this: 'These games, all games, are played on a four-inch field between your ears,'" he said that night. "That's where it's at. You have to be able to deal with this, not only physically, but mentally. I have no doubt that Andrew Luck, the person we know that he is, he's going to come through this thing -- and he and I have had long talks about it -- not just as good as he was, but a better quarterback. When is the question. That timetable is gonna be more on, really, the football gods and Andrew's gut feeling on how he's feeling,"
The Colts honored Manning last month by unveiling a statue of him outside Lucas Oil Stadium on Oct. 7 and retiring his No. 18 jersey at halftime of their game against the San Francisco 49ers on Oct. 8.
Luck had surgery in January to repair a torn labrum in his right shoulder. He practiced multiple times last month before being shut down and given a cortisone shot. When the soreness in his shoulder didn't subside, Luck was placed on injured reserve by the Colts last week.
"The consensus from all the doctors is to continue rehab, to be patient and continue rehab," Colts general manager Chris Ballard said last week. "The shoulder is subjective and after a surgery everybody's different, everybody rehabs at a different way. So we're going to shut his throwing down and we're going to continue to rehab. We're going to continue hard rehab."
Luck originally injured his shoulder during the 2015 season in Week 3 at Tennessee. He missed two games because of his shoulder and then missed the final seven games of that season because of a lacerated kidney. Luck missed a game last season with a concussion and all of this season with the shoulder injury.
ESPN's Mike Wells contributed to this report.