The Hammers are currently in 11th place in the table, six points behind 10th-placed Stoke City with a game in hand.
West Ham can cut that gap to three points on Tuesday night when they travel to Arsenal, but Allardyce believes that his players will struggle to catch up with Stoke in the coming weeks.
"I don't think that we'll get a top-ten finish now," he told the club's official website. "I think the best we would try and get is to catch up with Stoke. Stoke have had a fantastic run recently and we've got to try and improve our position to get where they are."
West Ham have lost four of their last six games in the Premier League.