Poyet has been billed as one of the Hammers’ brightest young prospects, following a summer switch from Charlton Athletic, but he has found his chances limited this season.
The likes of Mark Noble, Kevin Nolan, Morgan Amalfitano, Cheikhou Kouyate and Alex Song have all, understandably, been ahead of Poyet in the club's pecking order, but Allardyce has a window to finally give the youngster his chance.
The 19-year-old has proved that he is capable of playing senior football while at Charlton and out on loan in the Championship this season at Huddersfield Town, and West Ham now need to see if he is ready to make the step-up to their own first-team.
Poyet has played just five times for the Hammers this season, and his last appearance came back in January.
Although he has been a regular in the under-21 squad since, this will not be doing much to aid his development and Allardyce needs to give him an opportunity.
West Ham's season appears to be virtually over. The London side are safe from relegation, but they will not be making it into the European spots.
Although there is an understandable desire to ensure that they stay in the top-half, the pressure on results is not as great as it once was and it means that the environment is ideal for youngsters to acclimatise to first-team football in.
As a result, Allardyce should bring Poyet into contention and see if he is capable of performing at the top level.
On his brief showings so far, the technically gifted midfielder has looked comfortable, but it will be a bigger test to prove whether he can do this over a prolonged period of time.
If he can, then West Ham have a player that can figure prominently in their future plans, but if he can't then the side can look to move him out on loan to help his progression again next term.
Without testing him they will not know the best way forward, and Allardyce should use the coming weeks to try someone new in his team.
Although the manager's future at the Boleyn Ground is far from certain, the experienced boss has the chance to leave a parting gift to the club by introducing Poyet and it is an opportunity he should not pass up.