In November, Moyes was brought in as a replacement for Slaven Bilic and the Scot's sole objective was to keep the Hammers in the Premier League.
West Ham ended the campaign in 13th place with Moyes picking up 33 points from 27 matches but according to Sky Sports News, it will not result in an extended deal for the 55-year-old.
The report claims that while West Ham were pleased with the job that Moyes has done at the London Stadium, the club want to appoint a boss who they believe is capable of taking the team higher up the standings.
Moyes is now out of contract with the East London outfit, but no official announcement has been made regarding his future at West Ham.
In total, Moyes has recorded nine wins and 10 draws from 31 games in all competitions.