The Spaniard could find that he is soon out of a job if results do not improve.
The Hammers had expected to contend for a European spot after a big summer spending spree.
According to The Guardian, senior figures at the club will give Lopetegui one more chance.
They have come up with a managerial shortlist should he be fired in the next two or three weeks.
Managers on their radar include ex-Borussia Dortmund boss and former West Ham assistant coach Edin Terzic, former Denmark boss Kasper Hjulmand, ex-Benfica and PSV Eindhoven manager Roger Schmidt, and Stuttgart's Sebastian Hoeness.
For his part, Hoeness stated recently that he will be staying at Stuttgart for the rest of the season.
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