According to Sky Sports journalist Keith Downie, the Hammers submitted a bid for the highly experienced shot-stopper but the Championship club have turned down the offer.
The Championship outfit are under no pressure to sell the 32-year-old keeper this season. Moyes wants to bring him in as competition for Lukasz Fabianski.
It will be interesting to see if the Hammers return with another offer for their former player. Randolph has done well in the Championship and he could prove to be a useful option for West Ham.
The Hammers have had a poor season so far and Moyes will be looking to turn it around. The former Everton manager took over from Manuel Pellegrini after a series of poor results.
The Chilean was heavily backed in the summer but he failed to deliver during the first half of the season.
Moyes will be looking to bring in his own players now and it remains to be seen whether the Hammers decide to wait until the summer to bring in new players.
Right now, bargain signings like Randolph would be ideal. Players like him will add depth to the side and give Moyes some much-needed options.