The upcoming fixture with bottom club Sunderland is deemed as crucial, with the West Ham owners anxious about leaving it for too long before making a change.
The club still regret their decision to leave Avram Grant at the helm for so long when results were not going their way, in a season that predictably ended in relegation, and are eager not to make the same mistake.
Saturday’s 4-1 defeat to Liverpool at Anfield means the Hammers are just five points clear of the league’s bottom club Sunderland – who they welcome to Upton Park next weekend.
The fixture may well hold the key to Allardyce’s future at Upton Park, with league games against Arsenal and Manchester United to come, as well as a fierce Capital One cup tie with London rivals Tottenham.
Allardyce has found himself under increasing pressure after a succession of poor performances and results.
The Hammers have struggled for goals all season, and have often played a system without a recognised centre forward on the pitch.
Captain Kevin Nolan has also come under fire for his recent performances, and added to that by picking up a red card for a nasty challenge on Jordan Henderson during Saturday’s game.
Former Chelsea boss Roberto Di Matteo has been linked with the job should Allardyce depart, while Hammers co-owner David Sullivan is believed to be close with former England boss Glenn Hoddle – although he has been out of management for six years.