Adrian made 150 first team appearances for West Ham since arriving in east London from Spanish club Real Betis in 2013.
"Andy and Adrian have both been great servants to West Ham throughout their time at the club and we thank them for their loyalty and commitment to the Claret and Blue cause," joint-chairmen David Sullivan and David Gold said in a statement on the club's website.
"Both have provided us and our fellow supporters with some unforgettable moments, whether they were sensational scissor kicks and unstoppable headers in Andy's case, or penalty saves and goals in Adrian's."
Former France international midfielder Samir Nasri, who played six times for West Ham after joining them in January following a doping ban, will also depart along with Spanish striker Toni Martinez.
West Ham finished the 2018-19 campaign in 10th place with 52 points from 38 matches.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Christian Radnedge)
