Despite a strong first half of the season West Ham have slipped to 11th in the Premier League going into the final game of the season.
A 2-1 home defeat to Everton confirmed the Hammers will miss out on European football next season and increased the pressure on Allardyce.
West Ham finish their campaign against Newcastle at St James' Park on Sunday (3pm), the club where Allardyce left acrimoniously in 2008.
And according to The Telegraph, the 60-year-old's departure at West Ham will be confirmed on Monday.
Hammers owners David Sullivan, David Gold and vice-chairman Karren Brady are said to be meeting Allardyce to deliver the news the club will not be retaining his services with his contract expiring this summer.
Former Liverpool and Chelsea boss Rafa Benitez is the frontrunner for the job, with West Ham looking at a new option to take them into the Olympic Stadium in the 2016/17 season.
Ex-Manchester United manager David Moyes - now at Real Sociedad - is also a contender as is Hammers favourite Slaven Bilic.