Hull City manager Steve Bruce and Sunderland boss Sam Allardyce have been interviewed about succeeding Roy Hodgson, who resigned after the shock last-16 defeat by Iceland at Euro 2016.
"We've consulted widely in the game and spoken to a handful of people," Martin Glenn told BBC sports editor Dan Roan.
"The new manager's got to be someone who can inspire people."
Glenn also said the new manager will need to "build resilience" in players so they are able to deal with criticism on social media and the pressures of an "intensely passionate" English media.
"The British press, like it or not, are probably the most intensely passionate about the game in the world and that has a spill-over effect," he said.
"The consequence of which is people probably play not to make a mistake, as opposed to play to win.
"So the new manager's got to be someone who can inspire people to get the best out of themselves, build resilience and unashamedly adopt the kind of psychological techniques that other sports and other football teams have done."
Both Hull and Sunderland have urged the FA to move quickly so they can plan for the new Premier League season.
Bournemouth manager Eddie Howe and USA coach Jurgen Klinsmann have also been linked with the vacant England role - and the Daily Telegraph claims a third "mystery candidate" has been interviewed.