The 24-year-old striker is currently on loan from Liverpool, but the Hammers boss admits new financial restrictions could prevent a permanent move to Upton Park.
The 24-year-old forward, who is currently on loan at Upton Park from Brendan Rodgers' side, has scored six goals in 18 Premier League appearances this term. Starting next season, Premier League clubs could be in danger of losing points and being fined after a rules to limit wage bills and cap the losses sides can make to 105 million pounds over three years were ratified last Thursday.
"The hardest thing is the overall package and the overall negotiation that needs to go on and make it is sustainable," Allardyce told reporters. "I point to financial restrictions being implemented next season that could blow the whole deal, in one go. Financially, you're restricted to be able to do it.
"In one fell swoop the financial restrictions mean we won't be able to sign Andy Carroll from Liverpool because it is too expensive, even if we wanted him, which we do.
"Someone will have a bigger and better budget than us probably but this is what is going to happen."