The Hammers are still deciding whether or not to give Allardyce a new contract, with his current deal expiring at the end of the season, and it is believed by the Italian website that they will go for Poyet should he depart.
West Ham supporters have expressed concerns over Allardyce's direct style of play, and if he does leave the club are likely to have a complete change of ethos.
Poyet's aesthetically pleasing brand of football is far more fitting with the club's traditional principles, and this could be a reason why the Hammers board would feel the Uruguayan could be a good choice.
If this rumour is to be believed, the worry for West Ham supporters is that Poyet's previous time in the Premier League was not particularly impressive.
The 46-year-old has just over a 30% win ratio with Sunderland and he would need to improve this record considerably at West Ham to keep his job.
If Poyet did move to the Boleyn Ground, he would link up with his son, Diego, who has been a regular for the Hammers' development side this season.