Moyes took charge of West Ham in November previous year, signing a six-month deal following Slaven Bilic's sacking, with the club in the relegation zone. "However, we have made the decision that, as part of our longer-term plan, we will be appointing a new manager this summer and will be commencing this process immediately".
Other options include Burnley's Sean Dyche, Huddersfield's David Wagner and Marco Silva, the former Hull and Watford manager.
The latter left Watford in January after he was initially linked with taking charge of Everton before Big Sam did, but the Hornets blocked the move and then Silva's side slumped drastically as he was sacked.
In a stunning development, David Moyes has departed from West Ham United after just six months in charge.
Moyes had to work in a hard atmosphere at the London Stadium, West Ham's unpopular new home since they left Upton Park in 2016.More news: Stardew Valley PS Vita release date confirmed
More news: National Basketball Association draft lottery: Reaction around the league, and what's next
More news: Aston Villa heading to Wembley after draw with Middlesbrough in Championship play-offs
Now the club is looking for a "high-caliber figure" to "take West Ham United forward in line with our ambitions". "I'd like to think it was David Moyes, I really hope so".
Supporters' unhappiness with West Ham's board has been evident since the club moved to London Stadium and those frustrations spilled over in unseemly fashion when some fans invaded the field and tried to storm the directors' box during a 3-0 home defeat against Burnley in March.
'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, ' said Sullivan.
Moyes' assistants Alan Irvine, Stuart Pearce and Billy McKinlay have also left with immediate effect.
Sullivan has reportedly held talks with Shakhtar Donetsk manager Paulo Fonseca.