The report states that both Real Sociedad's David Moyes and Napoli's Rafa Benitez will be targeted for the role at the club, with Allardyce's contract expiring in the summer.
They say the current Hammers boss is expected to meet with the club's hierarchy to confirm his departure. This comes after he said in a press conference prior to their defeat to Everton that he was expecting discussions over any potential new contract to take place at the end of the season.
Benitez is out of contract at Napoli in the summer and could be an option for the Hammers - though that will only become clear at the end of the Serie A season - while the report suggests that former Manchester United and Everton boss Moyes is their preferred option, but he has no plans to leave Real Sociedad after only joining the La Liga side earlier on in the season.
After an impressive start to the 2014/15 campaign, West Ham's season has really faltered since the turn of the year. They were in contention for the European places at the halfway stage, following an eye-catching opening, but their form in 2015 has been more akin to those scrapping for Premier League survival - and has led to them falling into the bottom half of the table.
With a move to the Olympic Stadium on the horizon in 2016, the club's hierarchy will want to see significant progress on the field in order to attract crowds upwards of 50,000, and it's hard to see them being satisfied without consistently good performances and top-half finishes at the very least.