Allardyce has already been busy since the end of last season, bringing in forward Mauro Zarate, left-back Aaron Cresswell and midfield duo Cheikhou Kouyate and Diego Poyet.
But the former Newcastle United manager wants to further strengthen his options ahead of West Ham's Premier League opener against Tottenham on August 16.
"I think there are three or four positions - it depends on the budget," Allardyce told the club's official website. "You would say there is still room for a young central defender.
"You would also say we need another left-back to go along with Aaron and I think a front-line player and the possibility of a midfield player as well."
"Like all the young players, we'll try him out in the pre-season," he added. "We may try him out, if he progresses well, in the early stages of the Capital One Cup if the opposition are from a lower division than ourselves.
"That system progressed us to a fantastic run last season to the semi-finals and we nearly made Wembley."
Meanwhile, West Ham freed up space in their squad by releasing midfielder Alou Diarra on Wednesday.
"West Ham United and Alou Diarra have reached a mutual agreement to terminate the midfielder's contract with immediate effect," West Ham posted on their official Twitter account.
