Carroll arrived at Upton Park last summer on a deal from Liverpool which dictated that he would spend the entire 2012/13 campaign in the capital.
He has struggled with injury and form at times with the Hammers, but has been firing on all cylinders again of late and netted a crucial double in a 3-1 victory over West Bromwich Albion on Saturday.
Allardyce had called on his strikers to start delivering prior to that fixture, and he has now revealed that he is hoping to wrap up a deal for Carroll when the transfer window re-opens.
He said: "Would I like him here next season? The answer is yes.
"But there are a lot of things that need to be done. We will look at that at the end of the season. There is always a possibility.
"Andy showed today that he is not just a threat in the air, but is very good on the ground as well. His volleyed goal was fantastic technique.
"In fact, we scored three fantastic goals. I was delighted with our display and it moves us closer to the 39 points that will make us safe."