It called the speculation "extremely damaging" in a statement released today on its website, and urged the Football Association (FA) to "respect the disruption that this process is causing."
But it did confirm that it had given permission for Allardyce to talk to the FA at his own request, suggesting both the FA and Allardyce are serious about discussing the England job.
Sunderland just avoided relegation under Allardyce last season, finishing 17th in the Premier League. The club added today that "Sam is very much key to our plans" and that "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."
The FA - the governing body which runs the national team - is on the hunt for a new England manager after Roy Hodgson abruptly quit following a humiliating 2-1 defeat to Iceland in Euro 2016. It was called England's worst loss ever by some, and Hodgson immediately announced his departure at the post-match press conference.
Allardyce is by far the favourite to take the job now, with Ladbrokes giving him odds of 1/3 and William Hill 2/5. The bookies' next favourite, Jurgen Klinsmann - who currently manages the United States - is a distant second with odds of 8/1.
There were rumours last week that Arsenal manager Arsene Wenger was at the top of the FA's list for the England job, but it seems that the talks did not progress - perhaps because, as ex-Arsenal player Thierry Henry suggested Wenger would always prefer league football over the international stage."
This is the press release from Sunderland in full:
The Football Association contacted Sunderland AFC to seek permission to speak with our manager as part of what was supposed to be a confidential discussion process with potential candidates for the position of England manager.
At Sam Allardyce's request, we agreed to this.
Sam is very much key to our plans. 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.
The ongoing speculation over Sam's position is extremely damaging to Sunderland AFC, particularly at this crucial time of the season and we urge the FA to respect the disruption that this process is causing and bring about a swift resolution to the matter.
England's next game is an important World Cup qualifier against Slovakia on September 4. The FA will want to appoint a new manager long before then in order to implement any new systems and tactics in time.