Bruce, who joined Hull in 2012, led the club to swift return to England's top flight and recently said he would be staying a the KC Stadium following speculation in the media over his future.
"Manager Steve Bruce has held informal discussions regarding the England vacancy, although no official approach has been received from the FA," the club said on its website.
"We would hope to see the FA conclude their business quickly in order to avoid further speculation regarding Steve ahead of what is a season of huge importance for the Club following our return to the Premier League."
Former England manager Roy Hodgson quit after his team suffered a humiliating exit from Euro 2016 at the hands of Iceland in the last 16.
The FA has also approached Sunderland manager Sam Allardyce, who is being tipped as the favourite to succeed Hodgson by British media.
