The Hammers have opted not to renew the Scottish coach's short-term contract, which expired at the end of the season.
Hammers joint-chairman David Sullivan has revealed that the club is aiming to have a "high-calibre" successor to Moyes in place within the next 10 days.
"I would like to place on record my sincere thanks to David Moyes and his staff for achieving the target of keeping West Ham United in the Premier League," he said in a statement.
"Throughout his time here, David has carried himself with dignity and honesty and we have all found him to be a pleasure to work with. He, Alan, Stuart and Billy accepted the challenge and attacked it head on, turning around a difficult situation. They deserve great respect for the job they have done and they leave the club with our best wishes.
"When David and his team arrived, it was the wish of both parties that the focus be only on the six months until the end of the season, at which point a decision would be made with regards to the future. Having taken stock of the situation and reflected now the campaign is complete, we feel that it is right to move in a different direction.
"We are already taking steps to identify and recruit the manager we believe can take West Ham United forward in line with our ambitions. We aim to appoint a high-calibre figure who we feel will lead the club into an exciting future for our loyal supporters within the next 10 days."
Moyes took charge of West Ham in November last year, signing a six-month deal following Slaven Bilic's sacking, with the club in the relegation zone.
The former Manchester United and Everton boss guided West Ham to a 13th-place finish.