The Hoops had their initial bid for the 23-year-old striker rejected by Wolves, but are expected to go back with an increased offer within the next 36 hours.
If the Molineux club still refuse to budge, Celtic will move on to other signing targets already identified.
The confirmation Griffiths is the name in the frame has not met with universal approval among Celtic fans hoping for a bigger name to arrive in this window.
But, while Lennon concedes Griffiths is not a £10million player, he is confident he can be developed into a very effective goalscorer, capable of making his mark at this domestic level - and above.
This is based on studying Griffiths for the past 18 months, and Lennon cites impressive Celtic success stories as examples of how players can come good in the Hoops. He said: "People would have said when we brought Gary Hooper in that they hadn't heard of him. You hadn't heard of Wanyama, Izaguirre, players like that. People were saying: 'He can't do this and he can't do that.' They all did it. We believe, if Leigh comes here, he can play at this level comfortably.
"And, at his age, there is a good five or six years for him to improve and get to his peak.
"Scott McDonald came from Motherwell and scored a huge amount of goals and went on to score in Europe as well.
"I think Leigh can do the same. He is a good player and he will come here and make us better.
"I wouldn't say and haven't said he is better than what we have, but he will add to the squad and improve it. It's not just his goals. He can see a pass and his movement is very good. He has got physically stronger and has got pace as well."
Lennon added: "I think he can do a job for Celtic. And, speaking to a lot of people in the game whose opinion I trust, they are in agreement.
"He's not a £10m player, but I think he can turn into a very, very good acquisition for us."
However, having come unstuck when they tried to negotiate a loan move then the purchase of Kevin Doyle from Wolves last year, Lennon knows that prising Griffiths from the League One club will not be easy.
Wolves boss Kenny Jackett has included Griffiths in his squad for tonight's game at Oldham and insists the ball is now back in Celtic's court.
He said of the initial offer: "The bid was turned down straight away. Leigh has expressed his interest in Celtic publicly and, for a Scottish lad, the pull of a club of that size is a big one. But he knows the deal has to be right for us.
"He certainly hasn't closed the door on Wolves at all, and he'll be at Boundary Park."
The Hoops are not willing to risk keeping all of their eggs in one basket right to deadline day on Friday, and will be ready to switch to other names on their list.
Lennon confirmed: "There will come a step where enough is enough, and we move on. But I don't know when that will be. We have other irons in the fire.
"We don't have bids in for any other players at the minute, but we're hoping we can get some things done over the next couple of days."
Lennon confirmed the deal they are attempting to broker for Griffiths is cash only, with no other players involved.
The Hoops boss is only interested in strengthening his squad, and that means he will be turning a deaf ear to any calls about the likes of Virgil van Dijk, Fraser Forster and Emilio Izaguirre.
"We won't be encouraging any bids," insisted Lennon. "I'd like to make it as plain as possible that these players are not for sale. We don't need to sell - and we don't want to sell."
Lennon did indicate, however, that Georgios Samaras and Joe Ledley have been informed the club wants a definite answer by Friday on whether they will be signing new contracts.
Meanwhile, after his brief sub appearance against Hibs, Stefan Johansen is in line to make his first appearance at Parkhead in tomorrow's match against Kilmarnock.
And the other new Bhoy, Icelandic striker Holbert Fridjonsson, will take a significant step towards the first team when he plays in a reserve game today.
Lennon said: "Holmbert had a thigh injury, but there is a reserve game today and, hopefully, he will get some game time.
"This will do Holmbert the world of good.
"He is there or there abouts, but needs a bit of work before we will consider him for the first team."