The 31-year-old has struggled to make an impact at Liverpool since moving to the Merseyside outfit in 2010 and has only made three appearances this term.
Reds boss Brendan Rodgers has relied on the club’s youngsters during the opening 14 Premier League games, prompting out-of-favour Cole to be linked with a move away from Anfield.
Allardyce has already signed Liverpool striker Andy Carroll on a season-long loan deal, and the Hammers boss would be interested in a similar move for the England international.
“My chairmen extended as much money as they possibly could in August – and went beyond it,” said Allardyce.
“Whether they can find any more or not is something I will discuss with them in the near future.
“I think if there is a chance [of bringing Cole back to Upton Park], it would be more on the terms of a loan, than on a permanent deal for the player.”
Queens Park Rangers manager Harry Redknapp also hinted at a possible move for Cole on Friday, with the Hoops boss a keen admirer of the playmaker.
“I’ve always liked Joe,” said Redknapp.
“I signed him as an 11-year-old, no-one knows Joe better than me. I still think he’s got the talent and there’s a top player in there.
“I like him very much as a person as well as a player, but I’ve not made any enquiries for Joe or for anyone at the moment.”
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