The coach is contracted until after Euro 2016, but the governing body chief insists it is "not appropriate" to discuss the matter currently, while admitting Group D is a tough testFootball Association chairman Greg Dyke refused to talk about England manager Roy Hodgson's future after the World Cup, insisting it was "not appropriate" to discuss the matter before the tournament.
The former Fulham and Liverpool manager penned a four-year deal when he was appointed in 2012, taking him beyond Euro 2016.
Dyke refused to be drawn on a minimum requirement of progression for England in Brazil, but admitted their Group D rivals - Italy, Uruguay and Costa Rica - will prove a tough test.
"We wouldn't want to discuss what happens to him afterwards because this is the moment when things should be going well," he said. "So it's not appropriate to discuss that now. But I'm a big fan of Roy's.
"We would obviously hope to get out the group and then after that... It's a tough group. I think one of the reasons we are coming here with less expectation is because it's such a tough group.
"You compare with the others. If the French don't get out of their group I think they will be committing suicide. We are in a tough group and a lot of it is going to be about that first game."
Dyke also commented on the cut-throat gesture he made following the World Cup draw in December, insisting it was light-hearted and in reference to possible opponents and game location.
"The gesture was a joke," Dyke said. "We sat there and said there were two things we didn't want.
"We didn't want Italy and we didn't want Manaus, and we got both. It was a joke, and you sometimes have to accept there's a sense of humour in the world."