The Hammers have now gone six games without a win, and are in danger of ending their season on a real low note after such a great first half to the campaign.
Manager Allardyce is now under increasing pressure, and with his contract set to expire at the end of the season, he could well be entering his final weeks as West Ham boss.
With the West Ham hierarchy pushing to be more ambitious ahead of the club's move to the Olympic Stadium in 2016, big name managers are being linked with replacing Allardyce - but according to the Daily Mirror, it's Moyes who is leading the way right now.
The Scot is currently managing in Spain with Real Sociedad, and is rebuilding his reputation after a torrid spell as Manchester United boss last season.
After failing to replace Sir Alex Ferguson at Old Trafford, it's easy to forget that Moyes spent spent over 11 successful years as Everton manager, and would be a great choice for West Ham should they manage to lure him back to the Premier League - which the Mirror believe could be difficult given Moyes' desire to build a team with Sociedad.
The other hot candidate is Bilic, who played for West Ham as a defender during the 1990's. The Croatian, currently managing Besiktas, is a really charismatic manager who fans would really get behind, and he showed what he can do as a manager by knocking Liverpool out of the Europa League last month.
Of course, Allardyce would have to be let go for either of these managers to come in - but if the Mirror are correct, it seems that the Hammers are already planning for his exit.