Khan suffered a fifth-round defeat to his arch-rival Brook in what could be his final fight despite rumours of a rematch between the pair.
The former light welterweight world champion is believed to have made more than £5m for the fight and his wife could not resist a dig at Froch.
She tweeted: "I rarely ever say bad about anyone but I actually can't stand Carl Froch. Simply rude and annoying.
"Wonder if it upsets him that Khan made more money than him."
In response, Froch said Khan was 'shot' and he should call it a day.
"It's time to hang the gloves up. It's finished, it's done, he's got zero punch resistance.
"When you can't take a shot, it's dangerous. You know this in training because in sparring you get with shots.
"Amir Khan's either not sparring or he's sparring and he knows he's shot to bits.
"Either way he's got the answer that he shouldn't be stepping in the professional boxing ring again, ever.
"If he listens to this, I hope he takes my advice."
Froch added that he was 'laughing' at the comments his wife made.
"I know his wife wasn't very happy about my comments, but I think the less said about that the better."
When asked what he made of her tweet, Froch said: "I didn't make anything of it at all.
"All I did was crack up laughing and then go about my day."