But that will not be the case, and West Ham said assistants Alan Irvine, Stuart Pearce and Billy McKinlay had also departed.
Moyes succeeded Slaven Bilic in November, when West Ham were in the relegation zone, and guided the team to 13th place in the top flight.
Sullivan added on West Ham's website: "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."
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