The Sunday Mirror says any delay will inevitably lead to speculation about the manager's long-term future. His current deal expires in the summer and opinion around the club is widely divided.
Allardyce, who steered the Hammers back to the Premier League last May to crown his first campaign in charge, has always maintained he would walk rather than be pushed if he failed to deliver.
Yet, despite embarking on a lavish spending spree the Hammers have endured some winter wobbles on their return to top flight football.
Co-owners David Sullivan and David Gold are not prepared to open contract negotiations with Allardyce until the club are safe from relegation.