The latest bookmakers' odds have England Under-21 boss and former Middlesbrough manager Gareth Southgate as favourite to succeed Hodgson.
Odds are also being offered on Neville and Crystal Palace manager Alan Pardew (10/1); Bournemouth manager Eddie Howe and new Celtic boss Brendan Rodgers (16/1); Arsenal's Arsene Wenger (28/1) and Manchester United's Jose Mourinho (33/1).
Gareth Southgate no way for me
Odds are also being offered on Neville and Crystal Palace manager Alan Pardew (10/1); Bournemouth manager Eddie Howe and new Celtic boss Brendan Rodgers (16/1); Arsenal's Arsene Wenger (28/1) and Manchester United's Jose Mourinho (33/1).
Gareth Southgate no way for me