The Welsh midfielder, pictured, has been offered a new deal by the club, but has not given any indication on whether he is looking to stay in Glasgow.
Ledley has been a mainstay of Lennon's side since his arrival at the club and Lennon is keen to tie him up on a longer deal.
"Joe knows our position on it and the way he is playing we'd dearly love to keep him," said the Celtic manager. "He's not got back to us. It's not take it or leave it - there might be a bit of give and take one way or the other, but the bulk of the offer's there. Having spoken with him it's not a concern that it's rumbled on."
Meanwhile, Lennon will take a look at Hiroki Yamada, a 24-year-old Japanese midfielder, when the player arrives at Lennoxtown for a trial.
The Celtic boss is keen to supplement his squad in January and work has already begun on sourcing players to bolster the squad.
"We're looking to bring him over and have a look at him," said Lennon.
"John Park has watched him a couple of times now and we've seen some footage. It's always good to see him up against the rest of the players and see how he copes with it.
"He can play left side or off the striker. He looks okay, but we'll not have a real idea until we've seen him here.
"I don't think there would be any money involved - he's a free."
Huddersfield Town striker James Vaughan has also been looked at a couple of times by Lennon, although the Hoops boss has been told by manager Mark Robins the striker isn't for sale.
Lennon, though, believes that with no European football after Christmas he will have greater scope to embark on various scouting missions.
He said: "It's something we've tried to do a little bit more, get out and look at various players.
"I know we've been linked with Vaughan but there were other players who caught the eye as well. Mark's already said he's not for sale.
"I spoke with him before the game so he was aware we were coming down.
"It's not a major problem. He is on a list of potential targets but there are others who caught the eye as well.
"It's just John's judgement but you always get a better feel when you see players with your own eyes.
"We did it with Hooper and Kelvin and the other British ones. It's difficult to go abroad obviously but if there's a window when you can go down and watch a midweek game it helps.
"It's not been a luxury I've had for a while."