Allardyce may be on his way out of the club when his contract expires this summer, but the club must also be looking to bring in new players.
As reported on the Paper Talk segment on Sporting Life, the Mirror reported that the Hammers have made Randolph a summer target.
The 27-year-old has had spells with Charlton Athletic, Motherwell and now Birmingham City, and has two caps for the Republic of Ireland national team.
A return to London with a Premier League team could be appealing to Randolph, who has been a consistent performer for Birmingham since his 2013 move from Motherwell.
With Jussi Jaaskelainen now 40, West Ham may well need a backup to star goalkeeper Adrian, who has been nothing short of sensational this season.
Randolph is out of contract at the end of the season, and Birmingham manager Gary Rowett confirmed to the club's official website in March that he was hoping to open contract talks with the goalkeeper in a bid to keep him at St. Andrews - and now West Ham's alleged interest could scupper those plans.