Joshua established himself as the leading heavyweight in the world on Saturday night by adding 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 undefeated former world champion Fury - should the latter ever 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 he has remained a consistent presence on social media.
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."
@anthonyfjoshua challenge accepted. We will give the world 🌎 the biggest fight in a 500 years. I will play with u. You are a boxers dream.— TYSON2FASTFURY (@Tyson_Fury) April 29, 2017
Fury last year surrendered the world heavyweight titles that 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.