The French side's president Michel Seydoux said they could make the 30-year-old an "interesting offer" to keep him full-time on Wednesday.
But Dalglish knows of no negotiations between the clubs as yet aimed at making that happen, and said: "He has never spoken to me. I don't know anything about it."
He added: "I know they won 2-0 (against CSKA Moscow in the Champions League) and the lad hit the post and is playing well - but he's been playing well most of the time.
"It is great for Joe he is doing so well but beyond that I don't know of anything that has happened."
Cole has made an impression at Lille and Seydoux said on Wednesday: "It is not a certainty that he will leave us at the end of his contract.
"If he has a certain good feeling with Lille and we can make him an interesting offer this is something that we will look into."
Source: PA