Tosun continues to shine for the Turkish outfit, who currently sit top of Group G in the Champions League, and they will face a battle to keep hold in January.
Everton are in need of a new striker after losing the goals of Romelu Lukaku and Tosun, who has fired home eight goals so far this season, has emerged as a top target.
Besiktas now have a decision to make, cash in during the January transfer window when interest will be sky high, or wait until the end of the season.
Their resolve looks sure to be tested before then with Everton, Tottenham, Crystal Palace and German outfit Borussia Dortmund all said to be in the running for his signature.
The 26-year-old is also said to be keen on a move to the Premier League and could push for a switch if offers start to come in. Everton's number one job is to appoint a new manager but they are still keeping tabs on potential transfer targets in the meantime.