Hull boss Steve Bruce was interviewed by the Football Association on Monday about the England manager's job.
"It's got to be the prime job that any Englishman could ever want to have," Bruce, 55, told Sky Sports.
Hull said Bruce held "informal discussions" with the FA but said there had been "no official approach".
Bruce said he was "highly flattered" to be considered for the England job, adding: "I put my case across and let's hope it was successful."
He added he was up against "a big pal of mine" in Allardyce and said whoever gets the job needs to be able to stop players being afraid in tournaments.
England were knocked out of Euro 2016 at the last-16 stage and failed to make it out of the groups at the 2014 World Cup.
Bournemouth manager Eddie Howe and USA coach Jurgen Klinsmann have also been linked with the England role.
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Both Hull and Sunderland want the FA to move quickly on a new manager.
FA chief executive Martin Glenn is fronting a selection panel that also includes FA technical director Dan Ashworth and board member David Gill.
Bruce led the Tigers back into the Premier League last season after relegation to the Championship in 2015.
The former Manchester United captain and defender, who was never capped by England, said in June that he would be staying at Hull.
However, amid talk of a takeover at KC Stadium, he accepted he could be replaced when asked if he was certain of his position.