The 29-year-old has been frequently linked with a move away from the Premier League this summer, and the Turkish Super Lig has been touted as his most likely destination.
Spurs boss Andre Villas-Boas is ready to let Adebayor leave White Hart Lane, as he looks to line up at least two more attacking reinforcements before the close of the transfer window.
The Togolese international is yet to agree personal terms with Besiktas, though, despite their generous offer of a four-year deal, with wages of around £3.5million per season.
Adebayor is reportedly reluctant to take a pay-cut on his current £100,000-a-week salary, with two years still to run on his Tottenham contract.
"[Emmanuel] Adebayor is a very talented player, anyone would like him to be a part of their squad," Besiktas coach Slaven Bilic told Lig Tv.
"We are interested in Adebayor however, we have to also consider his financial demands."
The cash-strapped Black Eagles need to consider their finances carefully, particularly in light of their recent UEFA suspension, which restricts them from playing in European competition.
Having already had an appeal against their one-year ban turned down, Besiktas have now taken their case to the Court of Arbitration for Sport (CAS), and Adebayor might also wait for the outcome before making a final decision on a transfer.