The Newcastle United keeper has spent two separate stints with the Scottish Premier League outfit, and Lennon is keen to hatch out a contract for the 24-year-old.
"We are hoping to speak to his agent and Fraser himself in the next few weeks," said Lennon. "He knows we are very keen to get things tied up with him as he has had a fantastic two seasons here.
"He is making all the right noises at the minute so it is a case of getting things sorted out."
Forster has helped the Bhoys keep 20 clean sheets in 33 matches.