Defeat to Hearts at the weekend left the Hoops 10 points adrift of Old Firm rivals Rangers and languishing in third spot in the Clydesdale Bank Premier League.
In the aftermath of the result, Lennon was quoted as saying: "If results are not better soon, then I will have to look at my own position."
But there is no doubt in Sutton's mind that his former team-mate is the right man for the job and will haul Celtic back into contention in the title race.
He said: "Neil is his own man. The nature of the job is if results aren't going particularly well, especially at a club like Celtic, then things do change.
"But I think it is very early days to expect the talk of Neil leaving or whatever.
"There is a 10-point gap at the moment and it's bridgeable. There are 30-odd games left and Celtic have a few things in their favour.
"They are due major players back from injury. They are missing last season's player of the year Emilio Izaguirre, their captain Scott Brown and Beram Kayal - people who have a big influence.
"A couple of wins in quick succession for Celtic, and a slip-up from Rangers, and it will all turn again.
"Neil has the strength of character to do that and he is certainly the man for the job."
As far as Sutton is concerned, now is the time for the players to stand up and be counted and back their manager.
He said: "I know what Neil was like as a player, I know he expected the highest standards as a player and he will as a manager.
"The problem he has as a manager is that he's asking people to do things and, at the moment, they are just coming up a little bit short.
"There is a lot of pressure playing at Celtic and Rangers and when things don't go well the criticism is inevitable.
"It's not easy playing at big clubs like these but Neil is aware of what he has to do.
"It's about time people stood up - as Neil did as a player - and backed the manager and move the club forward.
"Neil is doing his job, he will get the players fired up. The players need to really get a grip and work it out between themselves.
"It's tough playing here (Celtic) but they should be wanting more out of each other and dragging more out of each other. If they do that, I'm sure they can bounce back."
Despite their position at the summit of the SPL with an unbeaten league record, Sutton claims Rangers are far from invincible.
He said: "I don't think Rangers are overly strong this season.
"They have had a fantastic start to the season league-wise but I still think they are fragile.
"If they lose a couple of players to injury, the likes of Nikica Jelavic, I think they will go through difficult times.
"Rangers aren't the strongest side they have ever been, far from it. That's just my opinion.
"I think they've got frailties. Celtic just need to get on a run and the sooner they do it, the better."
Sutton was speaking at the launch of his autobiography, Paradise and Beyond.