Gareth Southgate could be in line to succeed Roy Hodgson as manager on an interim basis in the wake of England's latest embarrassing early exit from a major tournament.

English football has embarked on another round of soul-searching after one of the most humiliating nights in the country's history saw them go out of Euro 2016 2-1 to Iceland on Monday.

Hodgson quit on the spot and Football Association chief executive Martin Glenn hinted on Tuesday that Southgate, who has been in charge of the England Under-21 side since 2013, could step up on a temporary basis.

Southgate, 45, who had an unremarkable spell as manager of Middlesbrough but is perhaps best known for missing a spot-kick in the Euro 96 semi-final penalty shoot-out defeat against Germany, is the strong 2-1 favourite with the British bookmakers.

Glenn Hoddle, a former England manager who lost his job over comments about disabled people, and USA coach Jurgen Klinsmann are next in the running with the bookies.

West Ham manager Slaven Bilic was among the names linked with the England vacancy -- often referred to as the "Impossible Job" -- but Hammers co-chairman David Gold said on Wednesday that the former Croatia coach would not be "the slightest bit interested".

"He told us at his interview for this position (last year) that his mission was to manage at the highest level in England -- the Premier League -- with the club he loves West Ham United," Gold told West Ham blog site www.claretandhugh.info.

"England is simply not part of Slaven's mission. He is a very, very loyal man. He is loyal to his backroom team and to the board.

"We have backed him 100 percent and he us -- our commitment to each other is total."

English football chiefs say they would be open to another foreign coach and Arsenal manager Arsene Wenger has been mentioned in English media as a possible target after his deal with the Gunners expires in a year.

FA chief executive Glenn hopes to have appointed a successor by the start of England's 2018 World Cup qualifying campaign in early September, but said that if not, "we have an interim plan in mind".

Asked on Tuesday if he was speaking about Southgate, he said: "We are not talking about names today, but it would be a pretty obvious one to pick."

He, FA technical director Dan Ashworth and FA vice-chairman David Gill, the former Manchester United director, will lead the search for Hodgson's replacement.

"We clearly need an inspirational manager who can harness the big resources that the English game has got, everything we have now got at St George's Park (the national training centre), to make us more resilient in tournaments," Glenn said.