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.
United States boss Jurgen Klinsmann and former England manager Glenn Hoddle have also been linked with the job, but Allardyce appears to be the frontrunner.
The former West Ham, Newcastle, Bolton and Blackburn boss won plaudits for keeping Sunderland in the Premier League last season. He has one year left on his contract.
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.
Allardyce and the FA
The England manager will be chosen by a three-man panel – Gill, FA technical director Dan Ashworth and chief executive Martin Glenn.
They want the next boss to be a strong-minded, tactically savvy manager who will build a clear team identity and help shape the team into a cohesive unit.
Allardyce was previously interviewed for the role in 2006 after Sven-Goran Eriksson left following that year’s World Cup.
Eriksson’s assistant Steve McClaren got the job instead, and England failed to qualify for the 2008 European Championship.
Here’s what Allardyce has since said about the England job:
“I should have got it and I really don’t know why I didn’t. It had to be political for me, rather than my credentials. Maybe my external look isn’t to everybody’s liking.
“It was the right time and the right job for me but not from the FA’s point of view. That job doesn’t come around too often.
“It seems foreign coaches are still all the craze for the top jobs and that is a great shame. I also think that Steve not being successful was a massive blow for British or English managers, because it has put us down a peg or two.”
2015 – from Allardyce’s autobiography
“I wanted to do a real knock-your-socks-off interview for the FA, so I put together a PowerPoint which looked at every single detail.
“There was nothing missing. Nobody but nobody was going to beat it. But then Brian Barwick, the chief executive, told me there were no PowerPoint facilities at the interview venue, so I had to print off hard copies for the panel. So much for the progressive FA.
“I should have got it and, as I’m a better manager now than I was then, I believe I should be in the running whenever it comes round again.
“That’s not vanity or being full of my own importance. My track record entitles me to be considered.”
“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.”