The 25-year-old has just had a stellar season for the Black Cats, hitting 26 goals as they won promotion back to the Championship. Such form saw the forward earn a call-up to the Scotland national team before making his debut this summer.
There is interest from the Premier League and English championship, while Rangers still view the player as a forward option as they look to add more depth to the position following Cedric Itten’s departure and the uncertainty around Alfredo Morelos' future.
Stewart still has one year remaining on the two-and-half-year deal he penned when signed from Ross County in January 2021.
As part of that deal, Sunderland have an option to extend the contract automatically for a further year, meaning that if the player doesn't sign a new deal he is still unlikely to walk away on a free in 12 months’ time.
The club’s sporting director Kristjaan Speakman confirmed in May their desire to reward Stewart for his goal scoring form.
"Ross has had an incredible season so for us, we’re always trying to protect players like that who have performed so well and naturally, you want to try and reward players.
"It gives inspiration to other people as well [to perform] and you want to keep that cycle going. You want that culture at the club."
Meanwhile, Aaron Ramsey's future remains unclear following his loan spell at Ibrox.
The Welshman has a year left to run on his Juventus contract and is looking for a move so he can play first-team football ahead of the World Cup in Qatar later this year.
He had been in talks to link up with his former Juve boss and Italian legend Andrea Pirlo who has taken over at Turkish top-flight side Fatih Karagümrük but it collapsed at the last minute.
Club president Suleyman Hurman said: "We practically reached the stage of an agreement with Ramsey but then we gave up."