National Under-21 coach Gareth Southgate is the early favorite of a not very long list of mooted candidates to replace Roy Hodgson as coach of England following its embarrassing exit at Euro 2016. Having flirted with foreign coaches such as Sven-Goran Eriksson and then autocratic Italian Fabio Capello before bringing in the vastly-experienced Hodgson, the Football Association (FA) is likely to go for an Englishman if the pundits are to be believed. But none of the small list of leading English contenders have won a major trophy, something which does not augur well for its hopes of rebuilding the morale of a side beaten 2-1 in such abject fashion by Iceland Monday. Southgate — perhaps best known for missing a spot-kick in the Euro 96 semi-final penalty shoot-out defeat against Germany at Wembley — restored some of his reputation with victory at the prestigious Toulon tournament in May having had a disappointing European Under-21 campaign last year. The 45-year-old — who has been in charge of the England Under-21 side since 2013 — has managed just one club, Middlesbrough. He kept it in the Premier League for two seasons before it was relegated in the 2008-09 campaign. Former Manchester United star Gary Neville also resigned following the Iceland defeat. Former England captain Alan Shearer, who said the team was "pretty pathetic" and "clueless" against Iceland, has indicated he would like a go but his sole experience at managerial level is an unhappy spell at his beloved Newcastle United. But Shearer, who said "he hasn't got a cat in hell's chance" of getting the England job, subsequently called for Southgate to work under Glenn Hoddle, a former England manager. Hoddle was in charge of England from 1996-1999, a spell that included the 1998 World Cup in France where the team reached the second round before losing on penalties to Argentina. But the former Tottenham Hotspur and England midfielder lost the post in bizarre fashion after the FA sacked him following a newspaper interview in which he suggested disabled people were being punished for "sins in a former life." Shearer, who played under Hoddle, 58, told BBC Radio Tuesday: "I would back Gareth Southgate if he was to go in there. But I would also look at getting experience around him like Glenn Hoddle. "Glenn was brilliant as a coach for England in my opinion, he has got so much to offer and is still young enough so I would totally back Gareth and Glenn." Veteran manager Harry Redknapp, considered for the job before Hodgson was appointed in 2012, went even further by telling talkSPORT radio: "Glenn is your man. Why is Gareth the favorite and more suited? "I think we have got the perfect man, Glenn fits the bill for me. "He said something, but that was a long time ago. He deserves another chance." Others in the frame include the combative Alan Pardew, who guided Crystal Palace to the FA Cup final last term, losing 2-1 to Manchester United in extra-time. The young hope is Eddie Howe, who has impressed at Bournemouth, the 38-year-old taking it into the Premier League for the first time in their history and retaining their status last season. But his inexperience at the top level may mark him out as an England manager of the future rather than Hodgson's immediate successor. Scotsman David Moyes and Northern Irishman Brendan Rodgers have also been mentioned, but the former has not flourished since early promise at Everton while Rodgers has just taken over the reins at Scottish champions Celtic. Perhaps the man that a few years ago England and the FA would have yearned for, Arsenal's long-serving Frenchman Arsene Wenger, still figures on the pundits' lists. But would he, at 66, really want the extraordinary pressure that the England job brings?