Glenn admitted overnight the FA were 'getting close' to appointed Hodgson's successor and insisted they wanted a man who could 'inspire' his squad.
The FA chief executive added: 'We're not after a short-term mercenary, someone just to do the job for a couple of years.
"I want someone to come in to the England role to really work with not just the senior team but to make sure all the great work with the Under-16s, 17s, 18s - look at how well the Under-19s are doing now - and to knit all that together.
"We want someone to do a great job for the England national team but as well make sure all the development teams are laddering up to something more effective."
Glenn later told the BBC: "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."
Sam Allardyce's managerial career
1991-92: Limerick (player-manager)
1994-96: Blackpool
1997-99: Notts County
1999-2007: Bolton Wanderers
2007-08: Newcastle United
2008-2010: Blackburn Rovers
2011-15: West Ham United
2015-: Sunderland