Saints are keen on achieving promotion this season and Russell Martin wants to strengthen their forward department.
Southampton boss Martin has his eyes set on Sunderland star Stewart, who is in the final year of his contract, and he is his number one target.
Saints submitted an initial bid of £5m plus add-ons for the 27-year-old forward, but Sunderland rejected it.
Now it has been claimed that Southampton are ready to table a fresh bid and are set to increase their initial offer for Stewart.
It is suggested that the St Mary’s outfit might submit a bid in the region of £8m for the Scottish international.
Last season, Stewart scored ten times for Sunderland in the Championship, but the majority of the campaign was marred by injury issues.
Stewart is yet to feature for Sunderland this season and he is currently nursing an Achilles tendon injury; Southampton though look prepared to gamble despite his injury.
