Moyes, who has spent the last four years at the London Stadium in his second spell at the club, has just 12 months remaining on his contract, and the likes of Brendan Rodgers, Graham Potter, Paulo Fonseca and Michael Carrick had all been linked with potentially succeeding the Scotsman before talks over his future resulted in the decision to stay.
However, one man who is set to bid a fond farewell this summer is midfielder Declan Rice, with chairman David Sullivan confirming that they have "promised" to sell the Englishman even though he still has two years remaining on his contract.
Out of the "three or four clubs" to have shown an interest in Rice, Arsenal are believed to be leading the race for his signature as things stand, while Manchester City, Manchester United and Bayern Munich are set to rivals the Gunners for the £100m-rated Hammer.
Meanwhile, West Ham quartet Vladimir Coufal, Manuel Lanzini, Aaron Cresswell and Angelo Ogbonna all face uncertain futures at the London Stadium as they are due to see their contracts expire this summer. Lanzini is the most likely to depart as things stand, while Ogbonna is hopeful that he will be offered a new deal.
The majority of West Ham's additions from last summer still have a point to prove, particularly Gianluca Scamacca, who struggled to make the desired impact in front of goal - scoring just three times in the league - before suffering a season-ending knee injury in March.
With both Michail Antonio and Danny Ings on the wrong side of 30, getting the best out of the 24-year-old Italian will be high on the agenda for whoever is in charge next season.
In terms of possible incomings, a replacement for Rice will be a top priority, and Southampton skipper James Ward-Prowse is one name to have been linked with the Hammers along with Fulham lynchpin Joao Palhinha.
Leicester City winger Harvey Barnes is another said to be on West Ham's radar, while signing a new central defender and left-back could also be on the cards, especially if both Ogbonna and Cresswell leave for free.