Fury, who has retired or threatened to retire on multiple occasions, is in training to face Derek Chisora for a third time at Tottenham on December 3.
The 34-year-old threw out a direct challenge to Joshua in September but negotiations between the two fighters' respective promotional teams eventually broke down.
Fury told the High Performance Podcast: "I don't think I can retire today. I need the Joshua fight. We have been trying to make that fight for years.
"It's the fight that people want to see. It's the fight that I want to see as a boxing fan."
Fury indicated he is likely to struggle to call time on his career given the constant prospect of new challenges, starting with a projected unification showdown with Oleksandr Usyk earlier next year.
A new generation of British rivals, headed by Joe Joyce and Daniel Dubois, are also beginning to stake their claims to a shot at the reigning WBC champion.
"For the last four or five years, there has been this three-headed monster: me, Wilder, Joshua," Fury continued.
"Joshua and Wilder have been slain, and I'm the last one standing. All of a sudden, you've got some new people coming up now - Joe Joyce, Daniel Dubois, Usyk's gate-crashed the party.
"Now there's a load of new blood that wasn't there five years ago and it's like, 'can you beat this person?'
"But I think it would be an absolute dying travesty if me and Joshua didn't fight in this era."
"I just don't think he ever wanted the fight. They're saying it was because of commercial rights, how can a sponsor dictate who you fight?"— Boxing on BT Sport ð¥ (@BTSportBoxing) November 15, 2022
"Fact of the matter, it's called 'lack of balls' in the trade!"@Tyson_Fury on why the fight with Anthony Joshua didn't happen... pic.twitter.com/hBrvTK9cDY