The Hammers lost 4-1 to Liverpool at the London Stadium in the league on Saturday evening and are now in danger of getting relegated to the Championship at the end of the season.
It has been reported that West Ham board have lost patience with Bilic and want to replace him with Moyes.
The former Celtic central defender is without a managerial job at the moment, having parted company with Sunderland at the end of the 2016-17 campaign.
The Black Cats got relegated to the Championship at the end of last season.
Moyes himself has suggested that he would be interested in taking charge of West Ham if there is a vacancy.
The Scotsman told beIN Sports when asked if he would be interested in the West Ham managerial job: “I am interested, yeah, but until that vacancy becomes available; at the moment it’s not available.”
“If it becomes available, yes, but at the moment it is not available,” added Moyes.