The 31-year-old regularly traded the number one jersey with loanee Joe Hart last season, but finished the campaign as West Ham's first-choice stopper.
Although his contract expires in 2019, Adrian - who also refused to rule out a return to former club Real Betis - believes that the Hammers should prove their faith in him with a contract extension.
"I'm waiting to hear [Manuel] Pellegrini's ideas, and from there will make a decision," the Spaniard is quoted by Football London as saying.
"In addition to Pellegrini's opinions, the contractual situation is also important - if the club want me they should show it with an extension. We have to iron out the rough edges with the president, so it remains to be seen, but West Ham is my second home.
"Both options (West Ham and Betis) are there, the first thing is to sit down with the club and see the situation. In the Premier League I have a good name, and here with Betis I also have a good image - once we know [West Ham]'s plans we will make a decision."
Adrian has been at West Ham since 2013 and has made 145 appearances for the club.