Sunderland have given the Football Association (FA) permission to speak to their manager Sam Allardyce about the position of England boss.
But the Premier League club say they want Allardyce to stay, insisting he is "very much key to our plans".
Sunderland claim speculation about Allardyce has been "extremely damaging" and have urged the FA to "bring about a swift resolution to the matter".
England are searching for a new boss after Roy Hodgson quit on 28 June.
He resigned after the national team were knocked out of Euro 2016 at the last-16 stage by Iceland.
Photographs that appeared to show Allardyce, 61, at the home of FA vice-chairman David Gill were published in the media in the past 24 hours.
Allardyce had been on tour with Sunderland in Austria until he returned home at the start of the week, reportedly on transfer business.
Sunderland said they agreed to let Allardyce speak with the FA as a "potential candidate" after he requested permission, but are upset discussions did not remain confidential.
"After what was an extremely challenging season, we are keen to see a period of stability, both on and off the field, and we want him to remain as manager of our football club," read a club statement issued on Wednesday.
Allardyce is not the only Premier League manager linked with England.
Arsenal's Arsene Wenger, Bournemouth's Eddie Howe and Hull City's Steve Bruce have all been touted as possible candidates.
So have United States boss Jurgen Klinsmann and former England manager Glenn Hoddle, but Allardyce now appears to be the frontrunner.
The former West Ham, Newcastle, Bolton and Blackburn boss won a lot of plaudits for keeping Sunderland in the Premier League last season.
England begin their 2018 World Cup qualifying campaign on 4 September, when they face Slovakia in Trnava.
The two sides met at Euro 2016, battling out a goalless draw in Group B.
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