The heavyweight champion produced a devastating knockout in the ninth round to retain his IBF, WBO and WBA belts but attention quickly turned to the possibility of a unification bout with Fury.
Joshua seemed reluctant to talk about his rival at first but fans roared with approval when asked if they’d like to see the two face off.
“Of course I want the challenge. It’s not about the opponent, it’s about the legacy and the belt.
“Whoever has got the belt, I would love to compete with them. If that is Tyson Fury, let it be Tyson Fury. It’s no big deal.
“It’s one fight at a time, picking them off one by one. That’s all it’s about for me. I’ve got to stay focused.”
‘The biggest fight in boxing’
Fury is the current WBC champion and took to social media to call out Joshua after the fight, predicting he’d stop the 31-year-old within three rounds.
“I want the fight. I want the fight next. I will knock him out inside three rounds,” he said in a video posted to Twitter.
The reaction from the 1,000 spectators allowed into Wembley Arena on Saturday showed there is a huge appetite for the fight.
Joshua’s promoter, Eddie Hearn, assured it would happen next year.
“Starting from tomorrow, we make the Tyson Fury fight straight away,” he told Sky Sports.
“It’s the only fight to be made in boxing. It’s the biggest fight in boxing. It’s the biggest fight in British boxing history.”
Joshua shook off the lockdown rust with a comfortable victory over his resilient opponent on Saturday.
After a cagey start, Pulev faced the count twice in the third round as Joshua connected with a barrage of shots.
The Bulgarian, who has now lost just twice in his career, continued to smile at Joshua despite being pummeled by the champion and deserves credit for taking the fight as far as he did.
But Joshua, who deployed his trademark uppercut to good effect throughout, eventually wore his opponent down and knocked him out in the ninth round with a devastating straight right hand.