Allardyce hopes to take Cole on-loan in January.
The Hammers boss said: “I think it will depend where we are and what we do in the next week or two, particularly with our injury list.
“Also on whether the chairman can find any money to support the team. Whether they can find any more or not is something I will discuss with them in the near future.
“But I think if there is a possibility it would be more on the loan front than a permanent deal.
“You would have thought Joe would be involved at Liverpool by now because of all their injury problems and lack of strength in depth. He doesn’t appear to have managed to do that.”