Joshua reclaimed the WBA, IBF, WBO and IBO heavyweight titles with a unanimous points win over Andy Ruiz Jr in Saudi Arabia on Saturday, avenging his shock knockout defeat to the Mexican in June.
Fellow Briton Tyson Fury, the former world heavyweight champion, is set for a February rematch with WBC belt-holder Deontay Wilder in Las Vegas. Their first fight in December 2018 ended in a draw.
Joshua's next test is expected to be against the Bulgarian Kubrat Pulev in a mandatory title defence in April or May.
"I'm not really interested in Pulev - you want to see Fury or Wilder," said Welshman Calzaghe, the former undefeated world super-middleweight champion.
"They are the three main heavyweights and they need to get in the ring to see who is the best.
"The fans need to see that because they get paid enough money to fight. One name, one champion.
"Fury and Joshua are both superstars, there is no doubt about that.
"But I'm a fighter and I wanted to be the best. It's about being the best in the division and building a legacy.
"Unless those two fight each other, I think it is going to be a disappointment.
"But they will have to fight sooner or later unless one of them retires, gets hurt or has a bad loss."
Wilder displayed his prodigious punching power again last month by knocking out Luis Ortiz for a 42nd career win.
The Fury draw is the only blemish on the American's record, but Calzaghe reckons it is the self-styled 'Gypsy King' who would emerge on top from a three-way battle for greatness.
"I like Fury and think he is the best in the division," the 47-year-old said.
"I thought Fury won every single round in the first fight against Wilder, except the one he was dropped in. Hopefully he can perform even better this time.
"What he's been through is remarkable. How he's come back from as low as he was, I think he deserves to be champion.
"It would be fantastic to see an all-British fight between AJ and Fury. You've got two big heavyweights and anything can happen when those two guys lands a punch on the chin.
"But I think Fury is the better boxer and I think he's got everything to outbox Joshua and frustrate him."
Calzaghe, however, gave credit to Joshua for his comeback win against Ruiz Jr.
The former Olympic champion produced a focused and disciplined back-foot performance to score a landslide points victory over his out-of-shape opponent.
"He fought the right fight in not mixing it with someone like Ruiz," Calzaghe said.
"You're not going to do that against someone who's knocked you down a few months before.
"It was up to Ruiz as the champion to close the gap like he did in the first fight and make him engage, but because he was overweight he didn't have the stamina."
