The east London club are in negotiations with Liverpool over a permanent deal for the striker, who has scored seven goals in 24 Premier League appearances this season
The 24-year-old forward spent the season on loan at Upton Park, where he scored seven goals in 24 Premier League appearances.
Liverpool boss Brendan Rodgers has yet to confirm the England international's future remains at Anfield following his £35 million arrival from Newcastle in January 2011, while Allardyce insists that West Ham must continue their pursuit of the player.
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"It's our main priority. We have to pursue that to the very end. We'll find out quickly whether it's a yes or no, so that will be the first priority. If we can start [the transfer market] with Andy Carroll that'll be a great achievement for us."
Carroll assisted two goals in Sunday's victory and Allardyce believes that he could build a team around the forward if they secure a permanent transfer, maintaining that the England man should not be judged on his record in front of goal, with the striker's other attributes vital for West Ham's progression.
"The young age that he is, he can only get more and more in terms of his playing prowess and then we can build around that with what we want to improve around next," the manager added. "Building the squad, not bigger but in terms of quality, is my priority.
"He's a player who is wanting to enjoy his football. He wants to play every game, he wants to enjoy himself when he goes out on the field. He wants to use his strengths and score as many goals as he can but also work hard for the team. There was a 60-yard chase back today and I think that says it all.
"He'd be disappointed he didn't score today but his all-round ability is showing more and more in every game he plays.
"It's not just about his aerial prowess in the box, it's his hold-up play, his quality on the ball, retaining possession, fighting for 50-50 balls that make life difficult for centre-backs. His all-round game is getting better and better."
Carroll himself was buoyant after the game, writing on Twitter: "Great game today buzzing for the skip [Kevin Nolan] getting a hat-trick and enjoyed every minute of being at west ham!!"