Leno is ready to leave Arsenal this summer after losing his first-choice status to Aaron Ramsdale last season.
USA international Matt Turner is joining Mikel Arteta's squad this summer to add further competition between the sticks.
Lazio keeper Thomas Strakosha has been targeted as he comes off contract this month.
However, the Daily Mail says Leno, 30, as he looks to leave the Gunners, is now being favoured by Fulham boss Marco Silva.
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