The Ukrainian international joined Besiktas last January however, struggled to settle in and failed to knock Tolga Zengin off his porch as first-choice keeper.
The Istanbul giants had been in the hunt for a new goalkeeper following a string of costly errors from first-choice keeper Zengin but he managed to keep his place in the team.
Everton are reported to be keeping close tabs on Boyko and will scout him over the European Championships this summer.
The Black Eagles are considering parting ways with Boyko as they continue efforts to sign Arsenal keeper David Ospina.
Besiktas are expected to hold talks with the Colombian international after the Copa America.
The Eagles won the Turkish Super Lig this season and will compete in the Champions League.
Boyko made just 11 appearances for Besiktas this season but does have another four-years remanning on his contract.