England under-21 coach Gareth Southgate has been linked by British media with the job, amongst a list that includes Arsenal manager Arsene Wenger, Bournemouth's Eddie Howe, Hull City's Steve Bruce and former England manager Glenn Hoddle.
"It's got to be somebody who really knows English football," Dyke, whose term ends in July, told British media.
"But there's loads of them now, more of them than there are English. The harder question is why anybody would want it.
"They get media pressure that no one else in football gets."
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However, David Bernstein, a former FA chairman, backed Sunderland manager Sam Allardyce to succeed Hodgson.
"I'm not saying we should have an English manager. But, of the English managers, I actually would go for Sam Allardyce," Bernstein said.
"He's a very powerful character. I think he's got the personality, the strength, he's a good technical manager, he's very experienced and he's someone who perhaps could imbue confidence."
England next play in a 2018 World Cup qualifier away to Slovakia on Sept. 4.
(Reporting by Ian Rodricks in Bengaluru; Editing by Greg Stutchbury)
