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.
"I'm ready, I'm keen. The next fight I want is Tyson Fury, for the undisputed championship of the world," said Joshua.
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.
Whyte was next scheduled to be in action on 21 November but a rematch with Alexander Povetkin was scuppered when the Russian tested positive for coronavirus.
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.
Fury will be facing Deontay Wilder next and may have to defend against WBC Mandatory challenger Dillian Whyte as well.
Heavyweight champions Anthony Joshua, Tyson Fury agree financial terms for two-fight deal, says promoter Eddie Hearn
Joshua holds the WBA, IBF, WBO and IBO belts, while Fury is the reigning WBC champion.
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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.
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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.