Moyes’ side were booed off the pitch by the home fans at the London Stadium following their 1-1 draw against Crystal Palace.
The Scotsman delivered West Ham’s first trophy in many years when they won the Europa Conference League last season.
The Hammers also reached the semi-finals of the Europa League the previous year but fans have been unhappy with the team’s football this season.
But it has been claimed that Moyes’ job is under no threat at the moment from the West Ham hierarchy.
The Hammers board are pleased with the progress the team have made under the Scotsman and the way they have performed in Europe.
West Ham have only sacked Slaven Bilic and Manuel Pellegrini midway through a season since 2010.
However, Moyes’ contract is up at the end of the season and the board will review his job towards the final months of the campaign.
It remains to be seen whether Moyes does well enough to earn a new contract from the Hammers.