Forster has spent the last two seasons on loan with the Bhoys and made 47 appearances last season as they reclaimed their position at the summit of the Scottish Premier League.
And his representative, Mark Curtis, has confirmed he is to meet with Celtic chief executive Pater Lawwell to finalise a permanent move to Celtic Park.
"I will call Peter Lawwell on Thursday and arrange a time to meet and try to finish things off - we need to," Curtis told the Scottish Sun.
"Fraser's been away but I spoke to him and know where we need to be. I will now arrange to meet Peter."
Queens Park Rangers have been credited with an interest in the 24-year-old, but Curtis insisted his client has his heart set on a return to the Scottish capital.
"Other clubs are not in his mind," Curtis added. "Does he want to stay at Celtic? Yes."
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