Newcastle are desperate to keep hold of Shelvey, but a bid of around £20 million would force them into considering selling the midfielder to West Ham.
Pellegrini is said to be a big fan of Shelvey, and he feels that the Newcastle man could really help him at the London Stadium.
West Ham are yet to make an approach to Newcastle, but Rafael Benitez could now face a battle to keep hold of the Englishman.
West Ham need to sell to fund a move for Shelvey, but they have plenty of fringe players up for grabs.
And the Mail suggest that West Ham could look to put Newcastle under pressure in the final stages of the transfer window.
If Shelvey was to move to West Ham he would be in competition with the likes of Pedro Obiang, Mark Noble and Declan Rice for a first-team place.