England could give the job to Southgate on a temporary basis while exploring other options.
England U21 manager Gareth Southgate during the press conference
Current manager Hodgson announced his resignation from English football's top job moments after the shock European Championship last-16 loss to Iceland, with Southgate quickly installed as the bookies' favourite to take the helm.
England's Joe Hart and other players after defeat to Iceland
Martin Glenn, the chief executive of the Football Association, has reportedly already spoken to the England under-21s boss, with a view to offering him the job on a temporary basis.
Roy Hodgson resigned as England boss shortly after the defeat to Iceland on Monday
A permanent tenure has not been ruled out, but Glenn is allegedly willing to ensure England bide their time in getting the right man in, meaning the former Middlesbrough boss could be given the job for up to a year.
England face Slovakia once more in September as part of their World Cup qualifying campaign
Other candidates reportedly in the frame include Arsene Wenger, who may well decide that the increasing pressure at the Emirates is enough to end a 20-year stay at Arsenal, while Leicester City's surprise Premier League title win has seen Claudio Ranieri's stock rise significantly.
But a long-term successor is unlikely to be appointed any time soon, leaving Southgate poised to take charge of the team's 2018 World Cup qualifying campaign, which begins with a trip to Slovakia on September 4th.