The Wales midfielder is out of contract in the summer and has been offered an improved contract by the Parkhead club but negotiations have reportedly stalled.
The former Cardiff City player, 26, was prevented from answering questions on the issue following the Hoops' 4-1 win over Ross County in Dingwall last Saturday - in which Ledley scored a brace - and he again sidestepped the issue on Thursday as he looked forward to winning his 50th cap against Finland, adding to the speculation that he could leave in the summer.
However, the Celtic boss, who expects this weekend to conclude the deal that will bring 20-year-old Icelandic striker Holmbert Aron Fridjonsson to Parkhead from Fram Reykjavik for a fee reported to be between £100,000 and £150,000, said: "Obviously Joe is with Wales at the minute and is mulling things over.
"We have made him an offer, Joe is aware of that as is his agent and we are waiting on them.
"He seems happy here, his performance last week will have done his confidence a world of good because he has been a wee bit curtailed this season by injuries.
"We know how good a players he is, that's why we have offered him a new deal.
"My gut feeling is, looking at his body language over the last week or so, I'd like to think he would stay on."
The Northern Irishman, speaking after the club's AGM at Celtic Park today, insisted that Fridjonsson should be pushing for a first-team place almost as soon as he arrives in Glasgow.
"The fee is agreed," said Lennon. "It is up to the player and his representative to agree personal terms.
"I would imagine we would get it done over the weekend.
"He is one for the present. The season is just finished in Iceland so he may need a little bit of break so we may not see him immediately but I'd like to think he would be making inroads around Christmas time.
"He is a good player for his age, a good size, we had him in for a week's trial and were hugely impressed by him.
"It is not a huge outlay but we think he is a player who will get better and at 20, he has very good potential.
"Technically he is very good, he is excellent in the air, he leads the line very well and, for the Scottish game, he likes the physical side of thing as well."