The Wales international, like Georgios Samaras, is in the last six months of his current deal.
Ledley showed why his manager is so keen to keep him by driving in from 23 yards for the only goal of the game against Partick Thistle, extending the Hoops' unbeaten league run to 19 games.
Lennon said: "I want Joe to stay. We had a conversation on Tuesday. He has intimated that he would like to stay and we are hoping that we can speak to his representatives this week and get that finalised.
"I thought he was our best player by a considerable margin. Joe has showed great mental strength, it must be a very unsettling period for him and for Georgios as well.
"I need a spine to stay together, the likes of Scott Brown, Ledley, Kris Commons, Samaras - if we can get him - and a few others, because they have been big players for us."
On Samaras, linked with a £1m move to Hull, Lennon said: "Again, we need to speak to his representatives and see how the land lies.
"I think he would like to stay but whether we can strike a deal or not is another thing. As far as I know there is interest from other clubs."
It was Celtic's third successive win by a single goal to keep them 11 points clear of second-placed Motherwell.
Lennon, though, admitted his side were flat and in need of their break in Turkey which will come after the game against St Mirren on Sunday.
"We lacked spark, particularly in the final third," he added. "There were some good bits of play but I and the supporters expect better.
"That's two our of the last three games we have been flat and it looked like things are catching up with us. We need this break.
"We looked heavy-legged and mentally tired. We also need an injection of new blood. There is no question that the squad needs freshening up and we will endeavour to do that.
Partick boss Alan Archibald claimed his side should have been awarded a penalty when Ledley tangled with Christie Elliott.
"I didn't know at the time he was in the box but I thought it was a foul and I didn't know why he didn't give it," Archibald said.
"But I know now, it was in the box and that is why he didn't give it here. I am not surprised by that. Anywhere else on the pitch it is a foul.
"It is one of those ones you would say was a soft penalty but it was definitely a push, it was clear to see."