The 30-year-old centre-back has dropped down the pecking order under Erik ten Hag at Manchester United.
And Maguire has been heavily linked to a move away from Old Trafford in the ongoing transfer window.
David Moyes is an admirer of the Manchester United star and is pushing for a move to sign the centre-back.
West Ham submitted a bid in the region of £20m but saw it rejected by the Red Devils, who are seeking £30m in order to let their big money signing leave Old Trafford.
The Hammers deemed the price tag set by Ten Hag’s side too high and decided to move on to other targets.
However, it has been claimed that Moyes is insisting on signing Maguire this summer from Manchester United.
Now it remains to be seen whether West Ham will submit an improved bid for the centre-back in the coming days.
