The Football Association is expected to appoint Gareth Southgate on an interim basis before sounding out the likes of Arsène Wenger and Brendan Rodgers as part of an exhaustive and global search to identify Roy Hodgson’s permanent successor as England manager.

Hodgson tendered his resignation in the wake of the humiliating defeat by Iceland on Monday that eliminated England from the European Championship, but he was persuaded to attend one final media briefing at the team’s base here while the squad flew home to Luton. The outgoing manager, the highest-paid coach at Euro 2016 on around £3.5m a year, was clearly uncomfortable and stressed three times, “I don’t really know why I am here,” having delivered a statement after the game in Nice the previous night.

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The 68-year-old felt “very raw and very fragile” as he apologised to the supporters who had witnessed England’s latest spluttering performance at a major finals, with the rather unconvincing nature of the proceedings exacerbated by the FA’s chief executive Martin Glenn, sitting at Hodgson’s side, twice stating he does not consider himself “a football expert”.

Glenn and the FA’s vice-chairman, David Gill, will work with the technical director, Dan Ashworth, to identify a new manager, with Southgate, now in charge of the Under-21s, likely to be in charge when the senior side play their first World Cup qualifier in Slovakia in nine weeks’ time.

Glenn suggested the FA had opted against drawing up succession plans before Euro 2016 “because we didn’t want to undermine [Hodgson]” but, in the wake of arguably the most embarrassing result in the national team’s 144-year history, is intent upon appointing “the best man for the job”. He, Ashworth and Gill will consider foreign candidates but would favour a manager who has experienced life in the Premier League. The FA would be prepared to wait for an ideal candidate to complete a contract elsewhere, with the interim overseeing the qualifiers for Russia 2018 in the meantime.

That has opened up the intriguing possibility of a pursuit of Wenger, who is entering the final year of his deal at Arsenal and of whom Ashworth recently said: “Has Arsène got a fantastic understanding of the Premier League, of English players, of the English media, of the expectations of England? Absolutely. So would you rule him out? Probably not. He might not want it.” That may well be the case, and the Frenchman’s instinct may be to seek a new long-term deal at the Emirates Stadium. Rodgers spoke with Ashworth – Glenn insisted the England job was not discussed – before signing a one-year rolling contract at Celtic and has plenty of admirers within the FA, with the title-winning Leicester City manager, Claudio Ranieri, also likely to be considered.

“As I said, it has got to be the best man or woman for the job,” Glenn said. “More likely a man, but it’s the best person for the job. I don’t think we are ruling out anything. An ideal mix is somebody who has had experience of the English game, ideally at a significant level. That is what you would look for. We clearly need an inspirational manager who can harness all the resources that the English game has got, everything we now have at St George’s Park, to make us more resilient in tournaments.

“I won’t be drawn [on specific names] but track records are clearly a very important part of any process. If there were the perfect English manager, you would pick them. I am not sure there is but we would take a good look and make a rational assessment. That is the brief: an inspirational manager and management team to get the best out of a squad which has got high potential.”

Facebook Twitter Pinterest Roy Hodgson and the FA chief executive Martin Glenn arrive at a press conference in which Hodgson said: ‘I don’t really know why I am here.’ Photograph: Dan Mullan/Getty Images

He added that Southgate, a highly respected figure within the FA but relegated with and sacked by Middlesbrough seven years ago and in charge as the Under-21s finished bottom of their group at last summer’s European Championship, would be “a pretty obvious one to pick” in terms of the initial interim. The former England centre-half did oversee victory in the Toulon tournament last month.

Asked if the governing body would be prepared to wait for a candidate to run down his contract, Glenn said: “It is a possibility, of course. If you said: ‘This person is an absolute shoo-in, can you wait?’ then we are well placed with an interim solution. Or you might get a few interim solutions – a few managers to come and help. It is such an important decision, we have got to get the right person. To wait a few months – if that is what we had to do – would be the right decision. But it’s a hypothetical.”

Ashworth intends to speak to the outgoing coach Gary Neville and a number of leading figures within the game when canvassing opinion over who should take the role, and will also consult senior players – Wayne Rooney, Joe Hart and James Milner are the only members of Hodgson’s party with more than 50 caps – in the current squad. Rooney had earlier issued a statement denying those same senior players had lost faith in Hodgson during Euro 2016.

The England captain had suggested he was “interested and excited to see who the new manager is” in the aftermath of the 2-1 defeat to Iceland. “He needs to put his stamp on the team, whoever he is,” said Rooney. “I know one thing: if I was the manager coming in I’d be very excited. It’s difficult to see now, but we do have a good squad. Whoever comes in will have the players to move us on from where we are and take us one step further.”

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Hodgson conceded he had been utterly unable to achieve that over four years in charge having overseen only three wins in 11 tournament matches. He defended the progress made, pointing to the number of younger players who have broken into the senior set-up, but made clear his instinct had been to stand behind his resignation statement of the previous evening rather than address his team’s failings for a second time.

“My statement last night was sufficient,” he said. “I’m no longer England manager, my time has been and gone. But I was told it was important for everybody I appeared. I suppose that’s partly because people are still smarting from our poor performance yesterday. Someone has to stand and take the slings and arrows that come with it. I have tried over four years to do the best job I could do. The results will show it wasn’t good enough because the best job you can do is win a tournament. We haven’t come close to it.

“As a result I expect that myself and the team will be criticised. But I certainly don’t feel that I need to be Uriah Heep-ish about it in any way. I think, personally, the team has got better. I personally think we have made progress. I personally think the team in 2016 is better than the team which reached the quarter-finals in 2012.

“But it’s a bad day for me. I am very raw, very fragile. The last thing on my mind is what England should be doing next. I had my chance. It didn’t work out. Now it’s someone else’s turn. But it’s a fact of life: one particularly bad game has caused a lot go damage to me personally, to the team and to the team going forward. They now have a major bridge to repair. If they’d played better last night, maybe it would not have needed repairing.”