The Spaniard was supposed to be second choice goalkeeper behind Joe Hart, however he was promoted to the starting 11 as Hart's form dropped, making 19 appearances in the Premier League last season.
According to Spanish newspaper Estadio Deportivo, the goalkeeper is now looking for reassurances over his future at the club as West Ham signed Fabianski from recently relegated Swansea, potentially putting Adrian back on the substitute bench.
The Spaniard only has one year remaining on his contract and if Hammers boss Manuel Pellegrini can't reassure the goalkeeper that he will be first choice, he may be playing elsewhere next season.
Adrian's former team, La Liga side Real Betis, are reportedly interested in bringing the 31-year-old back to Spain in an effort to improve their squad in preparation for their campaign in the Europa League next season.
The goalkeeper has been at the club since 2013 and has made over 100 appearances for the club, making him a long-serving member of the Hammers.
West Ham are linked with signing a number of people this summer, with Pelligrini reportedly interested in signing former Arsenal midfielder Jack Wilshere on a free transfer and have reportedly made a £20m offer for Sporting CP and Portugal midfielder Bruno Fernandes.