The 30-year-old joined the Hammers in a £15million deal from Liverpool in 2013 – and his £90,000-a-week wages made him one of the club’s highest earners.
Carroll played 14 times, managing only three Premier League starts, in his final season at the London Stadium.
Meanwhile, Adrian, who arrived on a free transfer from Real Betis in 2013, will need to find a new club.
The Spaniard made 150 appearances for West Ham, but lost his no.1 spot following the £7million signing of Lukasz Fabianski last summer.
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And former Manchester City star Samir Nasri departs after signing a short-term deal in December. He featured just six times.
Joint-chairmen David Sullivan and David Gold said: “Andy and Adrian have both been great servants to West Ham United throughout their time at the club and we thank them for their loyalty and commitment to the Claret and Blue cause.