Southampton agreed a fee with the Glasgow giants ahead of their defeat at the hands of Legia Warsaw on Wednesday, according to reports.
The Saints were understood to be waiting for the outcome of the tie, before they would be permitted to complete their swoop for the England shot-stopper, as had been agreed by all parties prior to the match.
Despite Celtic's reinstatement to the competition, Forster is keen to complete his move to Southampton ahead of the start of the new Premier League season and will undergo a medical in Southampton on Saturday morning.
Forster has been on the radar of Southampton for the last two years and the move will bring a final conclusion to their pursuit of the Newcastle United academy product.
Should all the formalities be completed in time, Forster could be presented to the Southampton fans at St. Mary's Stadium when they take on Bayer Leverkusen on Saturday evening.
The move will see Forster, 26, link up with his England goalkeeping coach Dave Watson.
Watson and Forster are believed to have a close working relationship, with the former the driving force behind a move for the Englishman - who will compete with Artur Boruc, who is entering the last year of his current contract at the age of 34.
Southampton hit a rocky patch last season when Boruc broke his hand, leading to a major slump in form as both Paulo Gazzaniga and Kelvin Davis failed to ably deputise.
Forster will be the latest addition as manager Ronald Koeman creates a new-look Southampton side, following a summer of upheaval.
The departure of Mauricio Pochettino came first, leading on to further exits in the form of Rickie Lambert, Adam Lallana, Dejan Lovren, Luke Shaw and Calum Chambers.
Koeman has so far recruited Ryan Bertrand, Saphir Taïder, Dusan Tadic and Graziano Pelle ahead of the new season, with Forster set to be his fifth new face.
More are expected to follow, with Southampton heavily linked to defenders Marcos Rojo and Florin Gardos in recent days, with bids placed for both players.