The apparent blunder is likely to spark accusations of sexism against the high-profile broadcasters.
A Sky Sports spokesman said the off-air remarks were "not acceptable".
The pair commented on Sian Massey's appointment for the Premier League game between Wolves and Liverpool on Saturday, when they believed their microphones were switched off.
They concurred that female officials "don't know the offside rule" with Keys adding: "I can guarantee you there'll be a big one today. Kenny (Liverpool manager Dalglish) will go potty."
Keys then remarked on comments made by West Ham vice-chairman Karren Brady in The Sun on Saturday morning about the level of sexism in football by saying: "See charming Karren Brady this morning complaining about sexism? Yeah. Do me a favour, love."
The audio was obtained by the Mail on Sunday.
Sky released a statement which read: "The comments are not acceptable. They were not made on air but we have spoken to Richard and Andy and told them our views and they have apologised and expressed their regret."
The Football Association gave their support to Massey, who crucially made a correct borderline call in the build-up to Liverpool's first goal at Molineux.
An FA statement read: "The FA has made real strides in encouraging both male and female match officials to enter the game at every level, and will continue to offer every encouragement to all officials within the football family to progress to the highest levels possible. We are proud to have some of the world's best match officials, both male and female."
Suggested For You
SUBSCRIBE AND FOLLOW
Get top stories and blog posts emailed to me each day. Newsletters may offer personalized content or advertisements.Learn more