Scotland striker Griffiths has not played since December when it was decided he should take time out of football to address personal and mental health issues.
The 28-year-old returned to work at Celtic’s Lennoxtown training centre last month and is currently stepping up his recovery programme in a bid to regain full fitness.
Griffiths was a spectator at Hampden on Sunday as Celtic defeated Aberdeen 3-0 in the Scottish Cup semi-final to remain on course for the completion of a third successive domestic treble.
But Celtic interim manager Lennon is keen to have Griffiths available for selection as soon as possible and feels he could yet feature in the cup final on 25 May.
“What have we got until then, six weeks?” said Lennon.
“Yeah, it is realistic for Leigh. But he’ll have to play and train for at least two weeks going into that final, so he then has a target of four weeks.
“He’s had targets since he came back to Lennoxtown. He is working away in the background and doing his double sessions.
“Again, he’s a player we could have been doing with over the piece as we have been working really with just one striker in Odsonne Edouard.
“That one striker has done brilliantly for me, but it would be good to have another one in the building.
“Leigh is getting there. It’s a slow burner, but we understand that as he’s been out for such a long period of time.
“It’s going to be difficult for him to get into the squad with the way the boys are playing There are a lot ahead of him and have earned the right to play in a cup final.
“I’m going to have to maybe disappoint a few as well, but they’ve been brilliant. I can’t believe we are talking about a treble treble, it’s phenomenal.
“There were a few disappointed not to start on Sunday and a few the same against Rangers a couple of weeks earlier. I get that, but I can only pick 11 and now we have even more competition with those boys who were out for a concerted period of time coming back to full fitness.”