Cleary, who coached the county’s minor A ladies to All-Ireland glory yesterday, has been strongly linked with the position vacated by Brian Cuthbert.
The two-time All-Ireland SFC medallist was interested in succeeding Conor Counihan in 2013 but the job went to Cuthbert.
Cleary is viewed as a leading contender again, having worked with many of the current Cork players at U21 level in 2007, when the Rebels were crowned All-Ireland champions.
Cleary then managed the U21s to All-Ireland victory in 2009 and played a key role as a consultant to Counihan during his reign as senior boss.
Cleary was also involved with his club Castlehaven when they won the Cork senior football title in 2013.
Cleary’s daughter Laura scored 1-2 in yesterday’s minor A decider in Thurles. And Cleary said after the final whistle: “My name is out there but I haven’t even thought about it and that’s being quite truthful.
“When it broke, I was on holidays and this was the focus today.
“I’ll see will I let my name go forward — it’s a huge, huge job and I’ll sit down in the next couple of days when I get a bit of time.
“It’s like having a second employment with no pay. It’s not something that you jump lightly into. In the next couple of days I’ll see how the land lies.”
Cleary feels too much can be made of the role of the manager.
“To me, it’s all about the players. The better players will come out at the end of the day — a manager can only do so much and too much has been talked about what a manager can do for a team.”
The fallout to Cork’s All-Ireland qualifier defeat to Kildare nine days ago has been severe, with Cuthbert standing down following an eight-point defeat.
Cleary said: “There’s been a lot going on in Cork for the last number of years. Brian put his name above the parapet.
“I think there’s a lot of unfair stuff there, stuff on social media for guys going out to do their best.
“When teams do well the manager is put up on a pedestal that he’s nearly kicking the ball over the bar.
“It’s gone a bit too far but that’s the way of the world. People get hurt by it and families get hurt by it.
“I don’t think it’s particularly fair for people that going out to give their best to an amateur organisation.”