Moyes has suffered a miserable campaign with his side sitting 13th having won just four games all season.
Scoring goals has been the major concern and Moyes starts the year without top scorer Tim Cahill who has headed to Qatar to join Australia for the Asian Cup.
Hills make Liverpool's Roy Hodgson 6-4 favourite to go but he bought himself some time after Saturday's last-gasp win against Bolton.
Third favourite is Fulham's Mark Hughes at 10-3 while West Ham's Avram Grant is out to 5-1 after picking up eight points in his last four games and climbing off the foot to 15th.
Chelsea's Carlo Ancelotti is 5-1 with Mick McCarthy 11-2 and Gerard Houllier out to 6-1 after his side's fantastic 3-3 draw at Chelsea.
Meanwhile, Hills now go odds on that all three promoted teams will survive this season. They offer 4-7 on Blackpool, Newcastle and WBA staying up and 100-1 on all three being relegated.
It's so tight at the top that none are safe yet so I'd give the 4-7 a miss. Just seven points separate the bottom seven and the sides I'd currently fear for are 1-2 Wolves, 11-10 Wigan and 4-1 Blackburn.
Other prices are 11-10 West Ham, 11-4 Birmingham and Fulham, 7-2 Blackpool, 4-1 WBA, 9-2
Aston Villa and 6-1 Newcastle.
To finish bottom William Hill go 9-4 Wolves, 3-1 West Ham, Wigan, 9-1 Birmingham, Fulham; 12-1 Blackpool, 16-1 Blackburn and West Brom.