Neto has spent the last three years as deputy to Marc-Andre ter Stegen, making just 21 appearances in all competitions.
During the last campaign, the 32-year-old was restricted to just three starts in La Liga, two of which came in the opening weeks of the season.
With no prospect of becoming first choice at Camp Nou and just a year remaining on his contract, it appears increasingly likely that he will allowed to move elsewhere before the end of the summer window.
Speaking to Bild, Joannis Koukoutrigas said: "Fulham is trying hard to get Bernd. Very professionally. This is also true for some other clubs and shows his status. But as I said, for me he is the number one at Arsenal and we are therefore also relaxed.
"He is a German national goalkeeper, has always performed well at Arsenal and is an absolute team player. For me personally, Bernd is also the clear number 1 at Arsenal. With all due respect to the competitors."
Despite Koukoutrigas's comments, Leno was restricted to just eight appearances in all competitions during 2021-22, just two of those outings coming since the turn of the year.
With just a year remaining on Leno's contract, Arsenal will want to cash in on his signature, and the acquisition of Neto for a lower fee would be regarded as good business by the Gunners hierarchy.
Although Neto may target a transfer to a club where he would become number one, the Brazilian may be promised Europa League football should he move to North London.
Former club Valencia - who have just appointed Gennaro Gattuso as their new head coach - are said to be monitoring the situation.
Villarreal are another team that are allegedly regarded as a potential destination for Neto, who may be given the chance to play in the Europa Conference League.