The Liverpool Echo understands that Everton director of football Steve Walsh flew to Istabul earlier this week to try to conclude a deal for the 26-year-old.
Tosun has scored 12 goals in 22 games for Beskitas this season, and the Toffees thought that they had their man with their offer of £25m, reports the Echo.
However, the Echo adds that Besiktas are hoping to use the interest from other clubs in Tosun, namely Italian giants Roma, to hike up his price by £5m.
Everton manager Sam Allardyce has made a new centre-forward a priority in January after his predecessor Ronald Koeman failed to replace Romelu Lukaku.
Wayne Rooney is Everton’s top scorer with 10 goals this season, but the Toffees have been reliant on either Oumar Niasse or youngster Dominic Calvert-Lewin leading the line.
Tosun has eight goals in 25 international caps for Turkey, so although Everton may be forced into paying a premium fee, they would be getting a striker with a good pedigree.