BBC Sport: Sunderland boss in talks over England manager's job
-Read more: Sam Allardyce: Sunderland boss in talks over England manager's jobSunderland 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.
Kevin Davies, who played under Allardyce at Bolton, thinks his former boss would be a good choice for England.
"Where we are now as a nation, I think he is probably the right man," said Davies, who earned one England cap during his career.
"Motivation-wise, he is fantastic, the best I have ever worked with. He knows how to get the best out of a player.
"He has a lot of other strengths, too. He surrounds himself with the best people in the best roles, which allows him to plan and get the best out of his squad."
Davies also thinks Allardyce will improve morale in the dressing room.
"The atmosphere he will create will be similar to Wales or Iceland... that spirit, that togetherness, along with the ability that we have a lacked," said Davies.
"I think he has always thrived on being the underdog and I think England are going to go into a lot of games as underdogs in the future, so it probably will suit the way England are at the moment. "
Allardyce is not the only Premier League manager linked with the England job. Arsenal's Arsene Wenger, Bournemouth's Eddie Howe and Hull City's Steve Bruce have all been touted as possible candidates.
United States boss Jurgen Klinsmann and former England manager Glenn Hoddle have also been linked with the job, but Allardyce is the frontrunner.