The 26-year-old has been one of Southampton’s primary transfer targets this summer as new manager Ronald Koeman rebuilds his squad after a raft of departures from St Mary’s.
Forster earned a place in England’s World Cup squad after impressing Roy Hodgson with a string of superb performances in the Champions League for Celtic.
Capped twice by his country, Forster was the only player not to feature as England failed to progress beyond the group stage in Brazil.
In February this year Forster set a new Scottish league record for consecutive shut-outs, passing former Aberdeen goalkeeper Bobby Clark’s achievement from 1971, by going 13 matches and 1,256 minutes without losing a goal.
Celtic signed the 6ft 7in keeper on a four-year deal from Newcastle for £2m in 2012, following two seasons on loan in Glasgow, and it is understood his former club will be due a percentage of any sell-on fee.