The Hammers are currently 10th in the league table with 39 points from 28 matches, 12 points behind fifth-placed Liverpool.
The London outfit have been struggling for form and for points in the past few weeks, but Allardyce is confident that West Ham can win five of their 10 remaining matches.
“I’m now looking at the final 10 matches of our season and I can’t see why we can’t win half of them, which would put us on 54 points,” the Englishman wrote in The London Evening Standard.
“What has happened in the last 24 hours makes things tougher, but I’m still hoping for another 15 points.”
Can West Ham United pick up 15 more points?
Claiming half of the final 10 league matches is ambitious from West Ham, especially when one considers that they have not won a league game since January 18.
The Hammers take on Arenal at the Emirates Stadium on Saturday and then confront Sunderland at home.
Next month, West Ham will face Leicester City (A), Stoke City (H), Manchester City (A) and Queens Park Rangers (A).
In the month of May, Allardyce’s team will take on Burnley (H), Aston Villa (A), Everton (H) and Newcastle United (A).
On paper, games against Sunderland, Leicester, QPR and Burnley are winnable, and West Ham do have the ability to draw three more matches.