Joshua added the WBA belt to his IBF title with a dramatic 11th-round stoppage of Klitschko in front of an estimated 90,000 crowd at Wembley Stadium.
The defeat was Klitschko's second in succession, having been beaten on points by Fury in November 2015, and an all-British fight between Joshua and Morecambe’s undefeated former world champion - should the latter return to boxing - could prove even bigger.
"(Tyson) Fury where you at, baby?" Joshua said while still in the ring.
"I love fighting. Tyson Fury, I know he's been talking, I want to give 90,000 a chance (to see us), I just want to fight."
Fury has been inactive since his defeat of Klitschko having most recently struggled with depression, but has made plenty of noise on social media in the run up to the fight having returned to training in Marbella.
Following Joshua's comments in the wake of Saturday's win, Fury wrote on Twitter: "Challenge accepted. We will give the world the biggest fight in a 500 years. I will play with you. You are a boxer's dream."
In another tweet, he added: "Well done Anthony Joshua. Good fight, You had life and death with Klitschko and I played with the guy. Let's dance."
Fury vacated the world heavyweight titles he won by beating Klitschko in an effort to focus on his mental health problems.
The 28-year-old has won all 25 of his professional fights so far, 18 by knockout.
Joshua, meanwhile, now boasts a perfect 19-0 record following his win over Klitschko, with all of the 27-year-old's victories coming inside the distance.
Fury has hinted he could return on the undercard of the WBO world middleweight champion Billy Joe Saunders’ title defence against Avtandil Khurtsidze on July 8.
The pair are training together in Spain but Fury would require clearance from the BBBofC to make his comeback and remains the subject of a UK Anti-Doping investigation.
Another option for Joshua is a rematch with Klitschko, the Ukrainian great saying post-fight that he would consider his future but would only fight on against the London 2012 Olympic gold medalist.
"I fought the better Wladimir Klitschko, not the complacent one, the guy who realised his mistakes and wanted to put it right," Joshua said.
"I can only improve from this, if I don't then I'm a silly man. It'll be interesting to see what the next two to three years hold in my career, building on this evening.
"He's been very dominant. He took a loss against Tyson Fury. I kept hearing the 'obsession and the passion' to get victory. He came to show what he was about, that he still had it. I knew it was going to be tough for him, because I wanted to do the same.
"Maybe against anyone else in the division he may have come out on top. I have the ultimate respect for what he's achieved inside and outside of the ring.
“I wouldn't mind fighting him again; if he wants a rematch no problem."