The Hammers had been in a relegation battle until their third from final fixture against Wigan in April, where a 3-2 victory effectively guaranteed their safety.
But a month earlier, Sullivan criticised Zola's team following their 3-1 defeat by Wolves which he described as "pathetic".
Zola added through a statement from the League Managers Association: "Despite what has been a very difficult period for me, I have thoroughly enjoyed my time at West Ham.
"I would like to sincerely thank the players and my staff for their continued support, hard work and commitment.
"I extremely proud of their efforts, over a long and difficult season, in retaining West Ham's place in the Premier League.
"West Ham is a great club, with fantastic supporters and I wish the club every success in the future. The termination process of my contract is now in the hands of the LMA and I therefore will be making no further comment at this stage."
LMA chief executive Richard Bevan praised Zola for guiding West Ham during "a period of significant uncertainty".
Brooking laments Zola loss but looks to future
"Clear communication and a supportive framework at the top of a football club are essential for stability and delivering results," said Bevan.
"Gianfranco has had to manage his team through a period of significant uncertainty at the club and has shown dedication, resilience and commitment to his players, staff and the fans.
"Without a doubt Gianfranco is a talented young manager and I am sure that in the fullness of time he will be back working in the game."
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