The same report also claims that Tottenham and their London rivals West Ham are also set to make a bid for the prolific striker.
Spurs boss Mauricio Pochettino plans to strengthen his forward line and provide support for Harry Kane upfront.
Stoke and Spurs are being rivalled by West Ham who are currently the only club out of the three reported to have actually made a bid.
The Hammers have made a €7 million offer for the striker according to the Hurriyet and vice-chairwoman Karren Brady will travel to Turkey to hold talks with Galatasaray.
West Ham boss Slaven Bilic also confirmed that the Turkish international striker is on his winter transfer list.
Bilic told Turkish sports newspaper Fanatik: “Burak Yilmaz is one of the players currently on our transfer list.”
Yilmaz has scored 81 times in 140 games for Galatasaray since joining on a €5 million move from Trabzonspor in 2012.