The England winger is available from Chelsea this summer and has been offered to Napoli.
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However, the Telegraph says Fulham are hoping to convince Sterling about remaining in west London for the new season.
Cottagers boss Marco Silva is eager to boost his options in the wide positions and is keen to speak with Sterling. Talks with Arsenal winger Reiss Nelson are also continuing about a new deal after last season's loan.
Sterling spent last season on-loan at Arsenal, where he managed just 13 appearances.