The Premier League club announced the end of Moyes' reign on Wednesday and declared, in an official statement, that they hope to appoint a 'high-calibre' successor within the next 10 days.
Moyes reached the end of his short-term contract with the club at the end of the season, but his stewardship was not universally popular among Hammers fans.
Joint-chairman David Sullivan said: "I would like to place on record my sincere thanks to David Moyes and his staff for achieving the target of keeping West Ham United in the Premier League."
Ex-Manchester City boss Pellegrini, now managing in China, and Shakhtar Donetsk coach Paulo Fonseca are the two leading contenders for the London Stadium hotseat according to the bookmakers.
Fonseca reportedly held a meeting at Sullivan's house on Monday as he ponders his next move after winning back-to-back Ukrainian titles.