Turkish newspaper Fanatik claims the Istanbul club have made the Spaniard their number one choice as they plan to part ways with on-loan Liverpool goalkeeper Loris Karius after an unsuccessful time for the German.
But Besiktas will seek a loan deal and not a permanent deal, which might not be to Fulham's liking.
The 31-year-old has been frozen out at Fulham this season after making just two Premier League appearances for the Cottagers - both in August.
Sergio Rico has been the club's number one keeper for the majority of the campaign and Marcus Bettinelli was the second choice until his injury.
Fabri made 32 starts for Besiktas in their Turkish title-winning season of 2016-17.
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