Ukrainian goalkeeper Denys Boyko signed for Besiktas from Dnipro Dnipropetrovsk in the winter transfer window, but could be packing his bags again soon, according to reports in the Turkish media.
Fotomac report Everton are interested in signing Boyco, who could cost them €5m.
Besiktas have also received an offer of €1.5m from Greek giants PAOK, but it remains to be seen if that would be considered.
Besiktas signed Boyko for €3.3m six months ago, but the goalkeeper had a poor start at the Turkish club, making individual errors.
However, it remains to be seen if Everton are genuinely interested in the 28-year-old, who made several mistakes in the very few games he played for the Turkish side, as they can easily do better.
Everton are expected to sign a goalkeeper this summer, after the departure of Tim Howard, and are likely to intensify their efforts with the appointment of Ronald Koeman.
The Merseyside club have also been linked to Marseille skipper Steve Mandanda, who has been in top form this season and would be a far better option.