Paul suggested that the former British super middleweight world champion should take on mixed martial artist Anderson Silva on the undercard of his clash with Fury.
The 26-year-old, who beat Silva in his sixth and most recent professional fight, also said that Froch, who will be inducted into the International Boxing Hall of Fame this year, could make a “career payday” by agreeing to the bout.
Reminding Paul that he bowed out of boxing with a victory over George Groves at Wembley Stadium, Froch has now responded by saying that the social media personality isn’t “good enough” to even engage him in conversation.
“Jake Paul has called me out on his social media, apparently he wants to fight me,” Froch said in a video posted on Twitter.
“Just in case you didn’t know, my last fight was at Wembley Stadium in front of 80,000 fans.
“I won’t be fighting the Mixed Martial Arts legend Anderson Silva on your undercard, I’ve got too much respect for him. And your punch – did that even land? Very dubious.
“But anyway 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, 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. You’ll never, ever have one of them, you’re not good enough.”
A second victory over Groves meant Froch concluded his career with 33 wins and two defeats, retiring as WBA (Unified) and IBF super middleweight champion.
Paul will face the first recognised boxer of his young career when he and Fury meet in Saudi Arabia on Sunday.
Two previous attempts to set up an encounter between the pair have fallen through.
Froch believes that Fury will put an end to an undefeated start to the former Vine creator’s combat sports career.
“You’re not a professional fighter, you’re a performing clown,” Froch emphasised.
“That’s all you are. So keep my name out of your mouth until you’ve done something in the game.
“I think this Sunday your world ends. If it doesn’t and you are stupid enough to take a fight with me, when this right hand lands on your chin, I will put you into f****** orbit and knock that stupid barnet off the top of your head.”