Stewart is still recovering from an injury he sustained in January and is not expected to be fit to play until September.
However, that has not stopped Southampton from trying to snare him away from Sunderland towards the end of the transfer window.
Russell Martin is a fan of the Scottish striker and Southampton failed with a bid worth £4m plus add-ons on Tuesday.
However, Southampton remain determined to sign the forward and have now gone in with an improved bid.
The Saints have tabled a bid worth £6m with another £1.5m in add-ons to try and convince Sunderland to sell the striker.
Southampton are now waiting for Sunderland to respond to their offer with three days left in the transfer window.
The Black Cats have had their own struggles in their attempts to sign a forward this summer.
It remains to be seen whether they are prepared to cash in on Stewart with little time left in the window to seek a replacement.
