Reports on Friday morning linked the Hoops with the 26-year-old Scotland international who is said to be available at around half the £12million he cost the Black Cats when he moved there from Wolves in 2012.
At his pre-match press conference ahead of the Scottish Premiership clash with St Mirren in Paisley on Sunday, when the Parkhead boss was asked if £6million was beyond Celtic, he replied: "No, but people are quoting £6million pounds, but he (Fletcher) was bought for £12million.
"I know transfer fees change over the years but he is a youngish player still so I would imagine it might not be £6million we are talking about.
"He is a player that I think most teams in Britain would like but affordability might be a problem for us though. Could we afford to spend £6million for one player? Yes, we could."
Asked if there had been contact between the clubs, the Northern Irishman said: "No, not at all. It is speculation. I don't know where this story has come from, I honestly don't.
"I don't know why I am sitting here five minutes into my press conference answering questions on him."
However, when asked if he would be interested if Fletcher was put to him, the former Celtic skipper said "yes" before addressing the possibility of any new signings breaking the club wage structure.
He said: "I don't know, you would have to ask the board that but if we are going to bring in a big-name player then wages will obviously have to come into mind when negotiating.
"I wouldn't imagine we would break it by a lot if we were going to do that."
Lennon, however, was again keen to stress his desire to get a new face as soon as possible.
"What we want to do is sign players for the future, for the second half of the season and obviously with the qualifiers in mind for next year," he said.
"I think the squad needs an injection of fresh blood and that is what we are trying to do.
"We are in a position where we can spend a bit of money. It is a position we probably haven't been in for quite a considerable time, a position as strong as this, and we will endeavour to bring quality in.
"That is what I want to do.
"It is very difficult in the climate we are in, we have to sell at times to make ends meet but we have done that this season so we are looking to forge ahead."
One player who has definitely joined the squad is Mo Bangura who returned to training following his loan spell with Swedish side Elfsborg, for whom he played against the Hoops in the Champions League qualifiers last year.
The striker, however, will not be involved against the Buddies with his future still seemingly up in the air.
Lennon said: "He is back training. The reason we let him go on loan was to get games, he has had that and we will get a look at him in the next few weeks and see how he goes.
"He might want a change of scenery. I spoke to him briefly today but it is good to see him back and he is an addition to the squad."
Anthony Stokes claims he is unconcerned about possible new signings, and indeed, insists increased competition for places only serves to spur players on.
The Republic of Ireland striker, who has only scored once since early last October, said: "It doesn't concern me whatsoever. It is standard at the club.
"Every transfer window there is rumours and speculation that we are going to bring in players and nine times out of 10 we do.
"If competition comes in you have to up your game and that is good for the club.
"I am disappointed with my goal return this season but on the other hand, it is not like I am missing chances galore.
"I haven't had many chances in the games. We all need to buck up a bit and play a little better football."