Saints boss Russell Martin has made Stewart his top striking target and the club are trying to strike a deal with Sunderland.
The club have now put in their second bid for Stewart, of £6m plus £1.5m in add-ons, after seeing a £4m plus add-ons proposal knocked back.
Sunderland, who are claimed to value Stewart at £10m, have not yet responded to Southampton’s new bid.
It is suggested though that there is confidence a deal will eventually be done for Stewart to move to the south coast.
Sunderland may want any move done quickly in order that they can spend the cash on new recruits before the transfer window closes.
Stewart, who is into the final year of his deal at Sunderland, is currently out injured.
Southampton though are confident that the striker will be fit and firing soon.
