The hunt is on for the Hammers' 15th full-time manager after widespread reports this week that Allardyce will leave the club when his contract expires in the summer.
The East Londoners have been linked with a host of top managers from across Europe after slipping from fourth at Christmas to 11th with just three wins in 2015.
Besiktas boss Slaven Bilic and Napoli manager Rafa Benitez are two of the names that have been regularly linked to the post.
But now according to a report in the Daily Star, Bilic is ready to hold talks with co-owners David Gold and David Sullivan.
The report claims close friends of the former Hammer insist he would consider a return to his old stomping ground this summer.
After other candidates such as David Moyes, Eddie Howe and Jurgen Klopp were ruled out of the running it appears to now be a straight shootout between Bilic and former Liverpool, Chelsea and Inter Milan boss Benitez.
The Spaniard has refused to commit his future to Napoli and is believed to be yearning for a return to the Premier League.
According to West Ham fans' website Claret & Hugh, he even received a touchline ban for next Monday’s Serie A match against Cesena after swearing following a recent draw against Parma.
A disciplinary statement published by the Lega Serie A on Tuesday said the former Valencia man “at the end of the match, while leaving the playing field, declared at the top of his voice ‘this is s**t Italian football for you'."
Bilic vs Benitez
It is difficult to gauge Bilic's achievements in Turkey as Besiktas, considered the second biggest club in the country, currently sit third in the Super Lig behind Fenerbahce and table-topping bitter rivals Galatasaray.
He seems a plausible choice, particularly since coming under fire from his club's famously vociferous fans after their exit from the Europa League this season. But he is an unknown quantity in the English top flight where his every move would be under intense scrutiny.
And what of Benitez? According to the London Evening Standard's Ken Dyer, 'West Ham would be a realistic option for the 54-year-old'.
His career has been laden with trophies and he added another two to the cabinet with Napoli in the form of the Coppa Italia and Supercoppa.
But a side which had become accustomed to battling for the Serie A title and Champions League places in recent years currently find themselves down in fourth with little hope of dining at Europe's top table next season unless they win the Europa League outright - they are tied 1-1 after the first leg of their semi-final against Dnipro.
The feeling among some Hammers fans is that, like Alan Curbishley before him, the Spaniard once described as the 'luckiest manager ever' by Sir Alex Ferguson is simply too dour and does not play attacking football home and away true to the club's roots.