The FA has begun the search for a replacement for Roy Hodgson, whose underwhelming four-year tenure was brought to an abrupt end by Monday’s humiliating Euro 2016 elimination by Iceland.
England Under-21 boss Gareth Southgate could take temporary charge while FA chief executive Martin Glenn, who this week conceded that he was “not a football expert”, canvasses opinion within the game.
Arsenal manager Arsene Wenger is believed to be among the FA’s targets, with Glenn pledging to hire the best candidate regardless of nationality, but Dyke believes there are other obstacles.
“It’s got to be somebody who really knows English football,” said the former BBC director general, who is leaving the FA next month.
“But there are loads of them now, more of them than there are English. The harder question is why anybody would want it.”
West Ham issue hands-off warning over Bilic
West Ham co-chairman David Gold, meanwhile, has warned the FA off moving for Slaven Bilic, whose Croatia side denied England a place at Euro 2008 and whose stock has risen after a good first season in east London.
“I don’t believe he would be the slightest bit interested in taking on England,” Gold said.
“England is simply not part of Slaven’s mission. We have backed him 100 per cent and he us; our commitment to each other is total.
"This is his club and he is on his way to great things at the Olympic Stadium. I don’t believe he will be tempted away by anything.”