Anthony Joshua called Dillian Whyte an "idiot" but said he was open to facing his fellow Brit once more if Deontay Wilder was unwilling to fight him.
Whyte (25-1) delivered an 11th-round knockout of Dereck Chisora on Saturday before quickly calling out Joshua.
Joshua, who is 22-0, defeated Whyte in December 2015 and the WBA (Super), IBF, IBO and WBO heavyweight champion wants to face WBC holder Deontay Wilder.
While the 29-year-old is willing to accept a bout with Whyte, that will only happen if Wilder – expected to have a rematch with Tyson Fury – is unwilling.
"Dillian is an idiot, you know," Joshua told Sky Sports.
"You know why? I showed him some respect. I just told him straight, 'I knocked you out clean when we fought', but I said, 'let's forget about that because we both developed so let's prove who's a better fighter now'.
But Whyte has issued a warning to Joshua - advising him that he won't sit around waiting while the unified champion attempts to secure a more lucrative name like Wilder or Fury for a tentative return date of April 13 at Wembley Stadium in London.
"I'm no backup dancer. I won't wait for him. He's the champion and the man I want to fight, but I won't wait," Whyte told Sky Sports News on Sunday morning while confirming that a fight in the USA is an option.
"I won't sit around and wait two or three years. I will do my thing. My career doesn't depend on him. He isn't the be-all and end-all.
"I want one of the top three but if they aren't available we'll pick someone else. There are some good young fighters, undefeated guys, and guys with wins and losses on their record.
"I won't wait for this Joshua-Wilder-Tyson Fury nonsense and become a pawn in their negotiations."