Former short-term boss Moyes is due for talks with the board on Sunday, says The Sun.
West Ham acted after defeat to Leicester City.
Pellegrini, 66, spent 585 days in charge but the Hammers have won just seven points at home this season, fewer than any other Prem side.
In total, they won 23 of 64 matches under the Chilean, giving him a win percentage of 35.9 per cent.
And now Moyes is expected to return to the London Stadium on an 18-month deal.
The Scot reigned between November 2017 and May 2018, securing the club's Premier League status.
The club will end the decade in 17th place in the Premier League, exactly where they began it.