Sunderland manager Sam Allardyce is the favourite for the job, having also been interviewed to succeed Roy Hodgson, who quit after England's dismal showing at Euro 2016 where they suffered an embarrassing defeat by Iceland in the last 16.
British media said the FA would like to make an appointment this week if possible and by next week at the latest.
"Who wouldn't (want to be England manager), if you're English?" Bruce told Sky Sports ahead of a pre-season friendly against Mansfield Town on Tuesday.
"It's got to be the one prime job any Englishman can ever want to have. I'm highly flattered and extremely grateful to even be considered," added the ex-Manchester United defender.
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"Any Englishman, worth their salt, would be really, really proud to be England manager. If anything happens, then let's see what the rest of the week brings and, if it does, happy days."
The FA want an English manager although they have spoken to United States' German head coach Juergen Klinsmann as well as Bournemouth's English coach Eddie Howe, said British media.
Asked how his interview went, Bruce, who has led Hull back into the top flight after relegation to the Championship in 2015, said: "You never know with an interview. I hope it went OK. I put my case across and let's hope I'm successful.
"But then again I'm up against a big pal of mine in Sam, and apparently there's some other candidates in the mix too.
"What I said in the interviews has to remain there. But we've got to take away the fear, especially at tournaments.
"We've only won one of our last seven games at two tournaments, and we've got some good players, especially young players. That's got to be the main thing," added Bruce, who was an accomplished central defender but never played for England.
(Writing by Ken Ferris; Editing by Peter Rutherford)
