The former Ireland international is currently the England U21 manager. He led England to victory over Spain in July's European U21 Championship final. Treacy said Carsley is "well positioned" to succeed Gareth Southgate as England senior manager.
"From the outside looking in, when you think of what class of manager we're going to be able to attract for this Irish team, I think Lee Carsley is brilliant," former Ireland international Treacy told Virgin Media.
"When you look at the success he's had with the English U21s, and he has his finger on the pulse of the underage system, which is what we need coming through to our Irish senior team.
"If we can get Carsley, I think that would be a great match. Personally, I do think his stock is a little bit too high, having done so well with the English U21s.
"I think he's really well positioned for Gareth Southgate's job in the next couple of years. Whether he might want to hang around for that, I'm not too sure. He's played for Ireland. He's obviously is a very proud Irishman. If he gets offered the job, it would be very difficult for him turn it down.
"I would just think that personally, maybe, his stock is a little bit too high and it could be a step backwards for him."
Another former Ireland international, Damien Delaney, feels now is the time to change manager.
"It is about whether we should make the change now or not," he said.
"It's almost certainly now a matter of when not if [Stephen Kenny's time as Ireland football manager ends]. Why wait?
"The air has gone out of the balloon is what I'm trying to say. You need someone to come in for the next three games, maybe get some results and lift the mood again and pique the interest.
"Also on top of that, it buys the FAI more time to get a replacement in because we don't play again after he next set of fixtures until March. That gives you a good six months to work through the interview process.
"I know there will be a list of candidates. There's a story circulating today in the Examiner that Lee Carsley has been approached. If you can get Lee Carsley before the next set of fixtures, absolutely get it in place. I think now it's a matter of when not if. You need to do it now.
"Also for Stephen's sake as well. His head is on the block and he knows that it's coming down the line..."
🗨️ "I personally think Carsley's stock is a little too high. I think he's really well positioned for Gareth Soughgate's job.
🗨️ "It could be a step back for him."
— Virgin Media Sport (@VMSportIE) September 12, 2023