It was reported last week that Fulham were close to agreeing a deal with the Turkish club, after letting David Button move to Brighton.
The 'keeper himself told Turkish media that personal terms had already been agreed.
"As you know the Premier League is a major league and one I am interested in joining. Negotiations between the two clubs are ongoing but there is a verbal agreement with Fulham. I want to thank Besiktas for my time here," he said.
Besiktas finished fourth in the Turkish Super Lig last season, with Fabri keeping 12 clean sheets.
He would add to Fulham's list of summer recruits, joining Jean Michael Seri and Maxime Le Marchand at Craven Cottage.