IBF and WBA champion Joshua defended his titles against Carlos Takam in Cardiff a fortnight ago with talk now turning to who is next for the undefeated Brit.
WBC title holder Deontay Wilder appears top of the list with WBO champ Joseph Parker also an option according to promoter Eddie Hearn with Fury out of action after being suspended from boxing after testing positive for a prohibited substance last year.
Despite his extended hiatus from the sport Fury, the last undisputed heavyweight champion, claims he would beat Joshua "inside eight rounds", a claim Joshua has now responded to.
"Get fit you fat f***" he wrote on Twitter after posting a video claiming he was "ready and waiting" for an offer from Wilder.
The 28-year-old's opening social media salvo - aimed at New Zealander Parker - was a GIF of Kermit the Frog drinking tea. The tweet read: ".joeboxerparker Sips tea, continues to read tweets & refreshes email.. Still No Offer."
Wilder was next on Joshua's hitlist, with the Watford-born fighter posting a YouTube video of a lengthy interview during which he urged the American to set up a title fight between the two. An accompanying tweet said: "A little word on the BronzeBomber fight. Ready & waiting for the offer"
Wilder, who beat Bermane Stiverne in the early hours of Sunday morning, has accused Hearn of deliberately delaying the negotiations as he eyes a blockbuster meeting in 2018.
“Eddie is ducking me more so than Joshua,” he told the BBC. “If you want to stay at home like a little girl, this king has no problem travelling to knock out the champion.”
Joshua had earlier denied that he is the one dragging his heels, insisting that Hearn is doing all he can to arrange the fight.
“I'm serious about becoming undisputed heavyweight champion of the world,” he said. “We're just making sure that a lot of these guys who are making noise will back up what they're saying.
“But if Wilder's not making an offer we'll do it on my terms. Present him with an offer, and see how they feel about that offer.
“That only goes to the champions: Deontay Wilder and Joseph Parker. It's the same situation to both of them.”