The 33-year-old ex-Juventus star has been a failure for the Catalan giants.
Neto is firmly no.2 behind Marc Andre ter Stegen, which means he is free to leave this summer.
According to Sport, Neto prefers a return to Italy, but may also be tempted by a Premier League move.
Fulham want to get a deal done in the coming days, as they want a top keeper in place for the new season.
The Cottagers have chased several keepers this summer, including Thomas Strakosha, Alphonse Areola, Nick Pope and Bernd Leno, but failed to sign any of them.
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