Forster spent last season on loan at Celtic Park and made a big impact as Neil Lennon's side won their ninth successive title.
The goalkeeper was also a crucial part of their Scottish League Cup win over Rangers in December, making several key saves to deny Alfredo Morelos.
The Bhoys want the stopper back on a permanent basis and are prepared to let Oliver Ntcham move in the opposite direction to get their man.
Celtic are not against letting Ntcham go, but will not let one of their star players depart for what they perceive to be less than his market value.
The Sun report that Southampton are keen on the Frenchman, and view him as a potential replacement for Danish midfielder Pierre-Emile Højbjerg.
However, Forster's contract is a stumbling block that may well dent Southampton's chances of seeing the switch go through.
Celtic would have to agree to paying more than the £15,000-per-week they had to when they loaned the keeper but the cash strapped side may struggle to do so.
Ntcham has scored four goals and has two assists this season after joining the Premiership club in 2017 from Manchester City.
The Frenchman has made 117 appearances for Celtic since, but struggled to hold down a first team place last season.
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Saints, meanwhile, face Watford today in the Premier League at Vicarage Road, with Sofiane Boufal and Hojbjerg both facing fitness tests.
Only Manchester City have scored in more Premier League away matches than Southampton this season with Ralph Hassenhutll's side having only failed to net at Burnley's Turf Moor and against Liverpool at Anfield.
They need one win to set a new club record of eight away victories in a single campaign and are yet to draw a league game in 2020, having won five and lost six.