Fury launched a tirade of abuse at Price and other top British fighters in a series of expletive-laden messages that included a hilarious typo which made him a laughing stock on the social networking site.
"Ill take your sole and give it the devil," Fury tweeted. He declined to explain whether he wished to remove the underside of one of Price's shoes, or steal his pricey fish dinner, in order to impress the Prince of Darkness.
The British Boxing Board of Control have launched an investigation into the rant.
Liverpudlian Price responded to that and other tweets - including "I'm gonna smash your face in u s***house scouse p***k I look in your eyes and c fear!" and "It's hard to be humble when your as great as I am! Ali said this 1960s Tyson fury saying it now!!!" - on Wednesday.
The British and Commonwealth champion, who destroyed Audley Harrison inside 82 seconds on Saturday, said: “He’s idiotic and giving the sport a bad name the way he’s running round like a playground bully.
“We are grown men and no one bullies anyone as far as I’m concerned. There is a genuine dislike for each other and that is going to make for a great fight.
“I say keep talking, because he’s building the fight. The things he’s saying aren’t bothering me in the slightest. It’s like dealing with a kid at school talking absolute nonsense.
“I’d take the fight this week but the reality is it’s more likely to happen next summer sometime. Let Tyson do his promotional work on Twitter or whatever else and I’ll let my fists do the talking.”
Fury’s promoter Mick Hennessy responded: “Tyson Fury is certainly not a playground bully in any way, shape or form.
“If David Price really genuinely wants to fight we will put our world title plans on hold and we will make the fight."
He added that Fury's Irish traveller background makes him a target for abuse and causes him to blow up on occasion.
After Price's win, his promoter Frank Maloney offered to put up £500,000 for him to fight Fury, who is the holder of the WBO Inter-Continental Heavyweight title.
Price will next fight 45-year-old former British, Commonwealth and European champion Matt Skelton on December 8.