Roy Hodgson resigned as England manager after his side suffered a shock 2-1 defeat by Iceland to crash out of Euro 2016 at the last-16 stage.
The 68-year-old had been in charge for four years after replacing Italian Fabio Capello but has won just three of 11 games in major tournament finals.
Iceland – with a population of just 330,000 – were among the lowest-ranked teams in France at 34 in the world.
Hodgson, who won 33 of his 56 games as England boss, would have been out of contract at the end of the tournament.
The former Liverpool and West Brom boss was due to discuss a possible contract renewal but Football Association chairman Greg Dyke said he would only stay on if England “do well” in France.
