Fury has consistently made jibes at Joshua's expense, claiming in January that he would knock out his fellow Brit within two rounds. Last week he said he had stopped training after becoming tired of waiting for the fight to be agreed.
Fury is optimistic the fight, which would be one of the biggest in British boxing history, will take place as soon as possible.
Plans for a "Battle of Britain" have accelerated after Joshua, 31, stopped Bulgarian veteran Kubrat Pulev in the ninth round at Wembley Arena on Saturday to retain all three of his belts.
Joshua has a 24-1 career record as he continued to prove his shock defeat to Andy Ruiz Jr in June last year was an aberration.
An agreement in principle for Anthony Joshua and WBC champion Tyson Fury to finally go head-to-head has been in place since earlier this year.
Former Olympic champion Joshua defends his IBF, WBA and WBO belts at Wembley Arena against the veteran Bulgarian in front of 1,000 fans, limited in number by coronavirus restrictions.
Joshua pointed a finger in Pulev's face as they exchanged words while beefy security guards kept the peace during their weigh-in for Saturday's bout.
A robust, hardened 39-year-old Bulgarian, Pulev will get the chance to blow a turbulent heavyweight division wide open again when he fights Joshua at Wembley Arena in London on Saturday as a mandatory challenger.
The 39-year-old Bulgarian, the mandatory challenger for Joshua's IBF title, is in London after their planned bout in June was postponed due to the coronavirus.
An agreement is in principle for Fury to fight Joshua twice in 2021 should Joshua come through his world title fight with Kubrat Pulev at Wembley on Saturday.
December's bout will be the first time Joshua has fought since outpointing Andy Ruiz Jr to regain his titles on 7 December, 2019.
Britain's Joshua was originally scheduled to fight Pulev on 20 June, with a second date in July also mooted amid the coronavirus pandemic.
Tyson Fury says both he and Anthony Joshua have to focus on individual bouts before before much-hyped title showdown
Tyson Fury said he was eager to give British boxing fans what they want by fighting Anthony Joshua for the undisputed world heavyweight title but he said the duo must focus on their immediate opponents to make such a match happen.
Anthony Joshua will fight only once this year but might take on Tyson Fury twice in 2021, says promoter Eddie Hearn
Joshua, the WBA, WBO and IBF champion, is due to fight Bulgarian next while Fury, the WBC champion, has a rematch due with American Deontay Wilder.
Coronavirus Outbreak: Kubrat Pulev pledges to donate half of earnings from Anthony Joshua bout to battle against ‘devilish virus’
The IBF’s mandatory challenger Kubrat Pulev said he will donate half of his purse from the world heavyweight title fight against Briton Anthony Joshua to help medical workers fight the “devilish” coronavirus.
Coronavirus Outbreak: Anthony Joshua vs Kubrat Pulev world heavyweight title bout postponed, confirm promoters
The world heavyweight title fight between Briton Anthony Joshua and the IBF’s mandatory challenger Kubrat Pulev of Bulgaria originally scheduled for 20 June has been postponed, promoters Matchroom Boxing said on Friday.
Coronavirus pandemic: Anthony Joshua's heavyweight title defence against Kubrat Pulev could be moved to July, hints promoter Eddie Hearn
Anthony Joshua’s world heavyweight title defence against Bulgarian Kubrat Pulev at Tottenham Hotspur’s stadium on 20 June could be postponed to July due to the coronavirus outbreak, promoter Eddie Hearn said on Tuesday.
Hot summer ahead for heavyweight division with Tyson Fury vs Deontay Wilder, Anthony Joshua vs Kubrat Pulev
Boxing's heavyweight division will be heating up even further this summer, highlighted by a third fight between Tyson Fury and Deontay Wilder and Anthony Joshua defending his belts against Kubrat Pulev in London
Briton Joshua, 30, defeated Mexican-American Andy Ruiz Jr in a December rematch in Saudi Arabia to win back his IBF, WBA, WBO and IBO world titles
California State Athletic Commission has lifted the suspension of Kubrat Pulev, imposed in May, after he kissed a reporter in an interview. She filed a complaint asserting the kiss was forced and unwelcome.