It was suggested late on Sunday night by the Scottish Sun that the offer for the twenty-six year old stood at around £7.5m, although sources close to the Bhoys claim that the offer was "derisory" and not close to the speculated fee.
Southampton are keen to add to their goalkeeping ranks after a period of injury for Artur Boruc last season led to weeks of uncertainty and inconsistent performances from deputies Kelvin Davis and Paulo Gazzaniga.
Last season the Polish stopper broke his hand and missed six weeks of action, seeing Mauricio Pochettino's side slide down the table in the meantime.
The Saints are keen not to enable the potential for a repeat this coming season, and Fraser Forster is thought to have been a long-term target for the south coast club.
Six foot seven inch Forster has a close working relationship with Southampton goalkeeper coach Dave Watson, who joined the club from Norwich City earlier this summer.
Watson is the current England goalkeeping coach and is understood to be a big fan of Forster, with a large influence on the selection of goalkeepers for England's World Cup campaign in Brazil.
While the initial offer has firmly been rejected by the Glasgow giants, Southampton are expected to make further offers as they attempt to secure the services of the former Newcastle United man.