Jones, 25, has been linked with a number of clubs in recent times, but he now looks set to move to the Britannia Stadium after the clubs agreed a fee reported to be in the region of £9 million.
A statement on the official Sunderland website read: "The Trinidad and Tobago international travelled to the Potteries this morning (Wednesday) for talks with Stoke boss Tony Pulis. He is undergoing a medical this afternoon."
The Stoke website added that Jones had travelled "for discussions with manager Tony Pulis and chief executive Tony Scholes with a view to completing a move from the Stadium of Light in a club record transfer".
Danny Higginbotham, who played alongside Jones at Sunderland, believes the striker would be a major asset.
"Kenwyne is a strong lad and I don't think I've ever seen anyone jump as high as Kenwyne," Higginbotham told The Sentinel. "He's very athletic and knows where the back of the net is.
"I think we signed for Sunderland on the same day - and he's certainly come on as a player since. He would be very good for us. The way we play would suit him well. He has that strength you need. I know what a handful he can be, with both his pace and his strength."