The Mirror says fans are turning on the Chilean after six games without a win and a string of poor performances, the most recent being Saturday's home defeat to Newcastle.
Pellegrini was brought to the club on £10million a year in May 2018. He has been given free reign to make his own signings and is in the second year of a three-year deal.
The club remain supportive but the initial positivity of his bringing in the man who lifted the title with Manchester City has stalled at the London Stadium.
Concerns have been expressed at Pellegrini's tactics and team selection after a promising start to the current campaign.