Hodgson quit after failing to land England into the latter stages of Euro 2016
Premier League bosses Sam Allardyce and Steve Bruce both want the England job, a report in Thursday's Sun newspaper (page 64) claims.
Allardyce led Sunderland to safety last season while Bruce steered Hull City back to the top flight via the Championship play-offs.
This was Bruce's second promotion with Hull and fourth overall as a manager
However, the pair are said to be desperate to succeed Roy Hodgson as Three Lions boss, and would be available.
Allardyce signed a two-year deal at the Stadium of Light in October 2015, but has a get-out clause should the national team job become available, according to the Sun.
Allardyce is said to have an England escape clause
Bruce, meanwhile, is on a rolling contract in East Yorkshire and is yet to commit to the Tigers despite last season's success.
Allardyce was interviewed as a potential replacement for Sven Goran Eriksson 10 years ago, but was overlooked in favour of Steve McClaren.
Crystal Palace boss Alan Pardew and his Bournemouth counterpart Eddie Howe both feature highly on the bookies' list of favourites, alongside West Ham United's Croatian head coach Slaven Bilic.
Could Bilic return to the international frame?
Hodgson resigned in the immediate aftermath of England's shock defeat to Iceland in the European Championship's last-16.