The 57-year-old manager believes that any move for the England international before the transfer window closes would only be on a loan deal, though, with funds tight at Upton Park
With the club on the brink of clinching the signature of Wolves winger Matt Jarvis for a fee believed to be in the region of £11 million, Allardyce believes that he would have little left in his budget for a permanent move for Carroll, but that he'd still happily take the player on loan before the close of the transfer window.
"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," Allardyce told reporters.
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"We would see if we could pursue it and clinch it, but if we only finish with Matt and nobody else I would be happy with the squad I've got.
"While it is not quite as big as I'd like it to be, I think it has got a lot of quality.
"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."
Allardyce admitted his concern at few of his strikers having Premier League experience, with only Carlton Cole having previously featured regularly in the English top flight.
He added: "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.