Moyes was only brought in as Slaven Bilic's successor in November, but he is leaving the London Stadium after the club confirmed his exit earlier today.
Sky Sports now report that talks have been held with Manuel Pellegrini, with West Ham looking to explore the possibility of appointing the Chilean as Moyes' replacement.
Pellegrini, 64, has one Premier League title to his name having clinched top spot with Manchester City in 2014, but he left the Etihad Stadium in 2016 to make way for Pep Guardiola.
Now in China with Hebei China Fortune, Pellegrini could be tempted to make a return to the Premier League this summer, and West Ham are hoping to fancies another stint in England.
However, it's claimed that Pellegrini's wage demands are a major stumbling block and could halt any West Ham bid, with talks ongoing as David Gold and David Sullivan pursuing a move.