Koeman revealed after Saturday's 1-0 pre-season friendly loss to Bayer Leverkusen that a deal appeared close for the 26-year-old Celtic stopper, with the clubs having reportedly agreed a fee of around £10million.
And later that evening the club's official Twitter page announced the deal was done.
Twice-capped England international Forster, who has also previously represented Newcastle, Norwich, Stockport and Bristol Rovers, has signed a four-year deal with the Premier League club.
This has been a tough few months for those connected with the south-coast club, with a host of players exiting St Mary's in the wake of manager Mauricio Pochettino's departure to Tottenham.
However, those exits have swelled the coffers and Saints made Forster their fifth summer signing on Saturday.
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