Leicester City are also said to be keen on his signature.
Chantome is now reportedly open to a Bordeaux exit in the summer and all three teams are said to be interested.
Chantome is out of contract this summer and Everton, Stoke and Leicester appear to have their eyes on grabbing a bargain.
The 28-year-old midfielder has been linked with Everton in the past, whilst a Paris Saint-Germain player, whilst the interest of Stoke and Leicester appears to be slightly fresher.
Everton are looking to improve on their squad – especially with the likes of Leon Osman, Darron Gibson and Steven Pienaar all out of contract at Goodison Park this summer.
Stoke are also set to be after some new players to push higher up the table next term whilst recently crowned champions Leicester have concerns over the futures of N’Golo Kante and Gokhan Inler.
Chantome, who has one cap for the French national team, has played 30 times for Bordeaux this season and scored two goals.