Moyes has been under a lot of pressure from the Hammers supporters for their poor style of play.
Per talkSPORT, the club may be ready to make a change, with Bayern Munich boss Thomas Tuchel a target.
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The German is leaving Bayern at the end of the season and does not have any big offers as yet.
The West Ham board and Moyes are both aware that a vocal majority of fans do not want him to stay.
West Ham take on Bayer Leverkusen in a Europa League quarter final second leg on Thursday.