The Republic of Ireland striker, who played under caretaker Lennon during his loan spell at Parkhead last season, believes the 38-year-old is the man to revive the club's fortunes.
Keane said: "I am sure from Neil's point of view, he would like it to happen, and I am hoping he does get the job because I was very, very impressed with what he had to offer and what he did on the pitch with the lads.
"He is very passionate about Celtic, there's no question about that.
"He loves the club and I am sure if anyone can turn the club around and get them back to where they were a few years ago, I am sure Neil can do that."
The Hoops have been linked with a series of more experienced men in recent weeks, although popular opinion seems to be firmly on Lennon's side, and Keane is hoping he gets his chance.
He said: "People keep talking about that, but then you look at someone like [Barcelona boss Pep] Guardiola in the same situation as Neil and what he has gone on to achieve now.
"He is only going to get experience if he gets the job and gets games under his belt. He can only do what he did at the end of the season and win games for Celtic.
"He certainly did that, apart from the (Scottish Cup) semi-final, but the league games, he won eight in a row, maybe.
"He has been brilliant and if you ask any of the lads or any of the supporters as well, they certainly want Neil to get the job."