The England international spent the 2012/13 campaign on loan at Upton Park from Liverpool.
He endured his fair share of injury troubles, but ended the season fully fit and firing on all cylinders.
Allardyce has regularly expressed a desire to retain Carroll's services, but has also aired concerns regarding the costs involved - with a sizeable fee and a hefty salary required to land the striker.
He is, however, prepared to keep pushing the issue as the Hammers start piecing together their summer recruitment plans.
Allardyce said: "Our first priority is signing Andy Carroll, if we can.
"We have nearly finished our planning for pre-season and the players have all had their training programmes given to them so when they come back on 1st July they are ready to go.
"Then it's about the recruitment of the other players who we want as quickly as possible. That will be the most difficult part of the summer."
Source: Newcastle United Mad