Arsenal reportedly want £3 million for the experienced keeper who still has another two-years remaining on his contract.
The Turkish sports newspaper claim that the Colombian international wants guarantees of first team as well as Champions League football.
Besiktas are able to offer both and a realistic title challenge as well as a lucrative salary.
The Black Eagles won the league title last season but had the worst defensive record in the top three and their biggest weakness was in goal.
Besiktas did sign Boyko from Dnipro in January but he failed to have the desired effect and the Istanbul giants are back in the race for a keeper this summer.