The ex-Manchester United manager gaffer stepped in to fill the void left by Slaven Bilic in November with the club teetering above the relegation zone.
The Standard claim Moyes was told by Sullivan in a phone call on Monday that the club were contemplating a change because the fans were not happy.
Sensing they might be able to get Benitez from Newcastle as relations appear to be hard between him and owner Mike Ashley over spending, West Ham now look certain to try again.
Reports this week said West Ham had held "productive" talks with Paulo Fonseca, the manager of Shakhtar Donetsk, while Sullivan remains a long-term admirer of Newcastle's Rafael Benitez.More news: United Nations mulls global war crimes probe of Gaza killings
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'All through his time right here, David has carried himself with dignity and honesty and we have now all discovered him to be a pleasure to work with, ' stated Sullivan.
Ferguson is stunned by West Ham choosing not to keep Moyes on and questioned what the Hammers' expectations are as he feels they are not even a top ten team.
Moyes' assistants Alan Irvine, Stuart Pearce and Billy McKinlay have also left with immediate effect.
But an eventual 13th-placed finish, thanks to seven points from their final three matches, was apparently not enough to earn the 55-year-old Scot a new deal.
It was announced of Wednesday that Moyes would not kept on as the club look to "move in a different direction" with the owners saying they want to appoint a "high-calibre" within the next 10 days.