"Southgate is believed to have already been asked whether he is prepared to take over on a temporary basis and has indicated that he is," the Daily Telegraph said.
A three-man Football Association (FA) panel has begun the job of finding a successor to Hodgson, who resigned in the wake of England's humiliating 2-1 defeat by Iceland in the last 16 of the European Championship on Monday.
FA chief executive Martin Glenn has suggested that England might wait to fill the post because it is "such an important decision", meaning it is unlikely they will have a new man permanently in place before their first World Cup qualifier away to Slovakia on Sept. 4.
British media said that opened the way for former England centre-half Southgate.
"The Football Association is prepared to make Gareth Southgate interim England manager for the start of their World Cup qualifying campaign," the BBC reported.
The prospect of a delay has intensified speculation that Frenchman Arsene Wenger could take the post in the long term.
Wenger is the final year of his contract with Arsenal and was reported to be open to the idea of talking to England's football authorities about the job, but the FA may be unwilling to wait a year for the 66-year-old who has always maintained that he would never break a contract.
Other names linked with the job include Bournemouth manager Eddie Howe, possibly in tandem with former England international Steven Gerrard, Hull City's Steve Bruce, former England manager Glenn Hoddle and Italian Claudio Ranieri, who won many admirers by steering Leicester City to their first Premier League title last month.
The FA would have to move quickly to secure Ranieri, who is expected to sign an improved contract with Leicester next month.
Glenn told a news conference on Tuesday: "We clearly need an inspirational manager who can harness all of the resources that the English game... has got, everything we have now got at (the national football centre) St George's Park, to make us more resilient in tournaments.
"That I think is the brief... it is for an inspirational manager and management team to get the best out of a squad which has got high potential.”
(Reporting by Neil Robinson; editing by Clare Fallon)
