The incident took place in the aftermath of last month's 0-0 Premier League draw at home to Burnley, when Moyes had been asked if he felt under additional pressure due to Chairman Ellis Short's attendance at the game.
The former Manchester United manager claimed he regrets what happened and has been in touch with the reporter, Vicki Sparks, to say sorry.
"It was in the heat of the moment," said Moyes. "I deeply regret the comments I made. It's not the person I am.
"I accept it was a mistake. I spoke to the BBC reporter who accepted my apology.
"Sunderland have known about it for two weeks. Sometimes these things happen in the heat of the moment.
"If you look at my character and personality it can't be further from the truth.
"This can happen. We are in an industry where sometimes you don't always answer things the way you want to. It is difficult when you are put in front of the camera so soon after the games."
As well as Moyes himself, the BBC and Sunderland have both confirmed the manager has apologised to the reporter.
Moyes was caught on camera saying after the interview: "Just getting a wee bit naughty at the end there so just watch yourself.
"You still might get a slap even though you're a woman."
A smiling Moyes then added: "Careful the next time you come in."
Moyes has come under fire from the UK's shadow sports minister Dr Rosena Allin-Khan, who called on the Football Association to deal with the incident.
She wrote on Twitter: "David Moyes cannot get away with these sexist threats - the FA must take action immediately."
Former England striker Gary Lineker, who works as a presenter for BBC Sport, also condemned Moyes.
Using the same platform, he said: "Moyes incident highlights a tendency for some managers to treat interviewers with utter disdain. Pressured job. Well rewarded. Inexcusable."