The Eagles are desperate to sign a targetman in the January transfer window after the injury to Christian Benteke that looks set to keep him out until well into the New Year.
Tosun, 26, has been in terrific form for Besiktas - where he scored 20 league goals to fire them to the Turkish title last season.
Palace fell short with a bid for the German-born forward in the summer and he has been impressing this season in the Champions League.
However, that could prove to be a problem if Besiktas, who have six points from their opening two matches, qualify for the knockout phase of the competition.
They could delay any sale of Tosun until the end of the season, which would mean Roy Hodgson will have to look elsewhere for a striker.
Newcastle United and Tottenham Hotspur are also tracking Tosun and posted scouts to Eskisehir on Friday for the World cup qualifier, which Iceland won 3-0.