Carroll is keeping West Ham waiting as he hopes for other offers to arrive.
"At this moment in time, 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 programmes given to them to take away to start training while they are relaxing at home or on their holidays," Allardyce told whufc.com.
"Then, it's about the recruitment of the other players who we want as quickly as we possibly can. That will be the most difficult part of this summer."