Sunderland braced to lose Sam Allardyce to England job
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.
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 the England job.
Arsenal's Arsene Wenger, Bournemouth's Eddie Howe and Hull City's Steve Bruce have all been touted as possible candidates.
Allardyce seemed to have missed his chance having been overlooked when Steve McClaren was given the job in 2006 and recently ruled himslef out saying before the Euros.
"That's gone. For me to be interested in the England job, Roy would probably have to leave at the end of the Euros.
"And would England be interested in me? They say they are looking for an English manager, but will they do it? You've got this, 'what's sexier?' element now, rather than how good you are at doing the job."