Sunderland manager Sam Allardyce has yet to be contacted by the Football Association about the England job.
The 61-year-old is currently in Austria, where the Black Cats are preparing for the new season.
USA boss Jurgen Klinsmann and Bournemouth manager Eddie Howe have also been linked with the job.
Hodgson’s successor will be chosen by a three-man panel of FA technical director Dan Ashworth, chief executive Martin Glenn and vice-chairman David Gill.
They have already spoken to former internationals – including Frank Lampard, Rio Ferdinand and Gary Lineker – as part of its review and have not ruled out an interim appointment, with World Cup 2018 qualifying beginning in September.
England Under-21s boss Gareth Southgate has already ruled himself out of the running.
Allardyce was interviewed for the England job in 2006 following the departure of Sven-Goran Eriksson, but the FA instead chose to appoint Steve McClaren, who had been the Swede’s assistant manager.
The former Bolton, Blackburn and West Ham boss signed a two-year contract with Sunderland in October before steering the club to Premier League safety after suffering only one defeat in their final 11 league matches of the season.