Reds manager Brendan Rodgers said he would be a ‘nut case’ to let Carroll go – but that has not deterred Allardyce.
The Hammers boss said: “Many things in the final week of the transfer window emerge and should it be the case that we would be able to do something then I think it would probably only be a loan.
“My problem is there may not be funds available for Andy now.
“If Andy became available or anybody became available and they wanted to join West Ham I would ask the owners. At the moment that is not the case and we just have to wait and see if and when that situation arises.
“If we play Carlton Cole, Nicky Maynard, Ricardo Vaz Te, (Modibo) Maiga – they are all players that play in the front three wide positions.
“What isn’t there is a proven Premier League goalscorer, apart from Carlton Cole who can get 10 or 11 goals a season. Nicky, Ricardo and Maiga are not proven so you hope it will be all right, but you cannot tell until you get eight or 10 games in.”
The Hammers agreed a loan fee with Liverpool this month with a view to a permanent deal, but the 23-year-old did not want to move to Upton Park.
England-capped Jarvis is likely to be involved at the weekend against Swansea provided the deal is completed in time and Allardyce did not hold back in his praise of the 26-year-old. “Certainly from an attacking sense this is one of the best final-third crossers in the Premier League,” he said.
“His stats are extremely good and he delivers in the business end of the field.
“When you are trying to create in the Premier League and score goals, he is one of those players who can certainly give defences a really hard time and that is what we are hoping he can do for us.”
Last season’s facts do seem to back up Allardyce’s claims, as Jarvis attempted the most crosses in the top flight and had the highest cross completion rate.
Those statistics are even more remarkable given the fact the former Gillingham man was playing in a side that finished bottom of the table and Allardyce said: “This is a player of dribbling skills, pace, trickery, that not only provides chances but also relieves pressure on your defence.”