The 24-year-old played the final match of his season-long loan in Sunday afternoon's 4-2 win over relegated Reading at Upton Park, where captain Kevin Nolan netted a hat-trick.
"It is our target to make it a permanent situation," Allardyce said. "Hopefully we can get somewhere in the near future to do that. It's our main priority. There are negotiations between the club and Liverpool."
He added: "We have to pursue it to the very end one way or another and we will find out as quick as we can if it is a yes or a no. If we can start with Andy Carroll that would be a great achievement for us as he can only get better and better.
"Then we build around that in terms of how we want to improve next year, with better quality.
"Andy is a player that is wanting to enjoy his football. He lives for his football and wants to play every game. He wants to use his strengths and score as many as he can - but he also works very hard for the team."
The Irons were in complete control against Reading after a mistake from a new England call-up, goalkeeper Alex McCarthy, gifted Nolan the opener on 23 minutes, with Ricardo Vaz Te sweeping home a second following more good work from Carroll.
The Royals responded with two quick goals at the start of the second half through Garath McCleary and substitute Adam Le Fondre. But the Hammers made sure of a top-10 finish when Nolan netted again and then completed his hat-trick with a header three minutes from time.
Reading's fate had long been sealed and they finish the season in 19th place. However, Royals boss Nigel Adkins maintains there are plenty of positives to take ahead of a return to life in the npower Championship.
"There is a good feelgood factor going forward for next season, which is important," said Adkins, who replaced the sacked Brian McDermott in March. "We have changed the way the club has played football - we are looking to be more in control with keeping the football for longer periods of time and that takes time to get everyone ready to go and do that."