Khan, who is currently on holiday in Dubai, said he was splitting up with his American wife, Faryal Makhdoom Khan, and suggested she had got together with Joshua.
Joshua, the reigning world heavyweight champion, and Makhdoom both responded by denying the accusations, with Makhdoom accusing Khan of being unfaithful.
“Faryal moved on quick,” Khan wrote on Twitter. “Always mentioned to me how much she wanted to be with another guy, from all people another boxer @anthonyfjoshua.”
In subsequent tweets, former light-welterweight champion Khan added, “Left my family and friends for this Faryal. I’m not hurt but another fighter. I’m making it public. You getting the divorce #Golddigger.
“Mans like Joshua can have my left overs!”
Makhdoom, a model, hit back, “Accusing other boxers just because your boxing career is finished. Man up amir.
“After all these years. I’ve always defending (sic) a cheat, always stuck around for him, always had his back. And I’m called a gold digger?”
Joshua responded by posting a clip of the song ‘It Wasn’t Me’ by Shaggy, in which the protagonist tries to dodge accusations of infidelity.
He added, “Bantz (banter) aside, I hope you guys can resolve your situation or this is a hack as we have never even met! Plus I like my women BBW (Big Beautiful Women) #ItWasntMe.”
Khan and his wife have a history of airing their dirty linen in public.
Last year Makhdoom, 26, took to another social media forum, snapchat, to blast Khan’s family, accusing them of abusing and bullying her.
Khan, 30, responded by telling his wife and family to try harder to get along.
But he and Makhdoom also appeared on British daytime television in January to discuss rumours about his infidelity and a sex tape he appeared in.
Meanwhile, Joshua has been ordered by the WBA to defend his title against Cuba’s Luis Ortiz.
The 27-year-old had been due to face Wladimir Klitschko in a rematch, before the Ukrainian retired on Thursday.
The WBA says Joshua and Ortiz have 30 days from August 3 to agree the fight.
Joshua, who beat Klitschko at Wembley in April, has also been told he must face IBF mandatory challenger Kubrat Pulev before December2 or risk being stripped of his belt.
There had been plans for Joshua to fight Klitschko in Las Vegas on November 11 and Ortiz’s manager, Jay Jimenez, has said his fighter could fill the void.
Jimenez told Boxingscene, “We are next. We have no problem fighting on that date of November 11th in Las Vegas. Our mandatory takes precedent over Pulev’s and they know that.”
Ortiz, 38, has 23 knockouts from 27 wins, while Pulev, 36, has one defeat in his 26 fights, with 13 stoppages.
Joshua holds two of the four major belts, with New Zealand’s Joseph Parker the WBO champion, while American Deontay Wilder has the WBC belt.