The retired British boxer was recently mocked by the YouTube star for not making money during his career and responded by threatening to send Paul into "orbit".
In a Twitter video, Froch said: "So, Jake Paul has called me out on his social media. Apparently he wants to fight me.
"Just in case you didn't know, my last fight was at Wembley Stadium in front of 80,000 fans.
"You've got a job to do against Tommy Fury. If by some miracle you manage to beat Tommy Fury, then I'll listen to you.
"I'll have a little chat with you and see what garbage comes out of your mouth. See what you've got to say.
"But until then, don't talk about me in the same league as you. I'm a four-time world champion. Two WBC titles there. You'll never, ever have one of them.
"You're not good enough. You're not a professional fighter. You're a performing clown. That's all you are."
Froch predicted that Paul will suffer defeat to Tommy Fury this weekend and has vowed to silence the social media star if they do come face-to-face.
He said: "Keep my name out of your mouth until you've done something in the game. I think this weekend, your world ends.
"But if it doesn't, and you are stupid enough to take a fight with me, when this right hand lands on your chin, it will put you into f****** orbit and knock that stupid barnet off the top of your head."