The club has released a statement on the matter
Sunderland say they will continue to support David Moyes as manager but that his comments to a reporter were "wholly unacceptable".
He has been asked by the Football Association to explain himself after telling the BBC's Vicki Sparks she might "get a slap" if she continued her line of questioning.
The Premier League's bottom club say "such actions are not condoned or excused in any way".
In a statement, the club said: “Sunderland AFC would like to re-iterate the actions taken following the Burnley game on March 18.
“The exchange between the manager and a BBC reporter was wholly unacceptable and such actions are not condoned or excused in any way.
“David recognised this immediately, proactively bringing the matter to the attention of the CEO and apologising to the reporter.
“The club also spoke with both a senior figure at the BBC and the reporter personally, expressing its profound regret over what had occurred.
“The matter was treated with the utmost seriousness from the outset and the swift and decisive action taken by the club and the manager at the time ensured that it was resolved to the satisfaction of the reporter and the BBC, which was the priority.
“With both the BBC and the reporter agreeing that appropriate action had been taken at the time, the club continues to fully support David in his role as manager of Sunderland AFC.”
Moyes apologised on Monday, saying: "I deeply regret the comments I made. It's certainly not the person I am and I accept it was a mistake. I've spoken to the BBC reporter who accepted my apology and hopefully we can now move on.
"The club have known about it for two weeks, I told Ellis [Short] about it two weeks ago. Sometimes these things happen in the heat of the moment.
"We're in an industry where you don't always answer things the way you want to. It can happen to managers young and old. It can be difficult being put in front of the cameras so soon after the games."