The report states that the Premier League club are prepared to table €4 million for the Ukrainian international.
Fotospor go onto claim that the Merseyside club are looking to replace Tim Howard in the summer.
Boyko has been tipped to leave Besiktas despite joining from Dnipro Dnipropetrovsk for €3.3 million in January. The former Dinamo Kiev keeper has failed to break into the first team and has made just two appearances for the Black Eagles.
Everton have endured a difficult season and will be looking to improve their squad over the summer transfer window. The Toffees are in 11th place with 41 points in 34 games.
Besiktas at the top of the Premier League and are currently three points ahead of their Istanbul rivals Fenerbahce.