Carroll joined the Hammers from Liverpool on a one-year loan deal last August after struggling to establish himself at Anfield following his club-record £35million switch from Newcastle in January 2011. The 24-year-old has been hampered by injury during his time at Upton Park but, since returning to fitness, he has hit form over the last few months.
And Allardyce has also seen enough to want Carroll back at West Ham next season, believing the powerful hitman can develop into a "very good player" at the club. "He has shown everybody what he can do and he has massive potential to grow into a very good player if he ends up back here with us," Allardyce told reporters.
"He's so willing to learn and become better that I think we could do well for each other, but there is a way to go before that might get sorted.
"It's about negotiation and both parties have to be happy that the negotiation cannot go any farther and it's a 'yes' or 'no'. We hope it's a 'yes' and we hope it is as quickly as possible."
He added: "I think that my reflection of Andy is exactly what I said from the start - as soon as we get him match-fit, the sooner you will see the goals.
"That match-fitness never came until his run after his second injury and I'm still wondering what he might have done this year had we got him off that one minute before he did his hamstring in his first game (against Fulham)."
Allardyce also confirmed the club will not be extending on-loan defender Emanuel Pogatetz's stay at Upton Park.
Austria international Pogatetz joined from German side Wolfsburg during January's transfer window on a loan deal until the end of the season. Pogatetz has made only a handful of appearances with the Hammers though, mainly as a substitute, and Allardyce has decided he will not be looking to retain the 30-year-old.
Allardyce said, according to the London club's Twitter account: "We've already said to Pogatetz thanks very much, he's been a fantastic professional but it won't be extended, and we'll decide on the others after the game."