Besiktas have accepted a £5.4m [€6m] offer from Fulham for Fabri according to Fanatik.
Per the source, the Black Eagles have already started to look for alternatives.
The Spanish keeper has been closely linked with a move this summer and it the latest report turns out to be true he will be joining the Premier League side.
The 30-year-old keeper shone in the Champions League last season, playing an important role in Besiktas cruising past the group stage as undefeated leaders.
The Super Lig outfit signed Fabri from Deportivo La Coruna in 2016 and he went onto lift the league title in his first season at the club.
Fulham have been busy in the transfer market this summer with head coach Slavisa Jokanovic already making a double transfer coup, landing Nice duo Jean-Michael Seri and Maxime Le Marchand earlier this week.
Fabri has just 12 months remaining on his contract and Besiktas have not ruled out the keeper leaving.
The Istanbul giants have not however, released a statement regarding the latest Fulham transfer story.
The experienced keeper has kept 26 clean sheets, conceding just 57 times in 66 appearances in the Super Lig since joining Besiktas.