The WBC champion kept hold of his belt after a controversial draw with the Gypsy King on Saturday.
Twice Wilder floored the Brit but was widely-thought to have been outboxed throughout the 12 round thriller at LA’s Staples Center.
Fury suggested Wilder would dodge a second fight with him, with Joshua’s promoter Eddie Hearn heading to the States to thrash out terms for a Wembley spectacular between the heavyweight title holders for April 13.
However Wilder seems desperate to get back in the ring with the 30-year-old Fury.
He said: “Fury deserves the rematch, it's only right.
“God alone knows how Tyson got up from that last round knockdown and maybe the count was slow but I'm not going to complain.
“Together we brought back the passion for the heavyweight division and with it boxing.
REF JUSTICE Tyson Fury reveals four demands referee gave him to avoid fight being stopped after brutal Deontay Wilder knockdown
“Tyson somehow survived and came back from hell. Some people thought he deserved to win, but some people thought I won and some people though it was a draw.
“Remember that on the night I was 50lbs lighter and a lot of his punches were pitty-pat. I was the aggressor and thought I won.
“But seriously I want the rematch before any other fight. No-one else. It's his right and it's good for boxing.”
And on Fury amazingly getting back to his fight in the final round after being seemingly KOed, Wilder admitted his amazement.
He said: “I saw his eyes roll slowly in the back of his head. Many people felt that should have been waved off. Nine out of 10 refs would have waved that off.”
Joshua has been vocal since the clash, claiming he is ready to face either man.
But Wilder says he has “finally come out of hiding” and is now “getting what he deserves” with the star looking set to miss out on a big-money showdown.
When asked about AJ and Hearn, Bronze Bomber said: “They thought they were the only rulers of the heavyweight division. They had four months of opportunity.
“Their egos got the best of them so let them continue fighting second-tier heavyweights while Fury and I continue making boxing great again. I do not want any other fight. No one else.”
And when quizzed on whether Joshua would be a bigger fight, Wilder remained adamant he wants to finish business with Fury first.
He added: “You thought the first fight with Fury was great, wow, wait for the second. As for Joshua, maybe me and Tyson will do a trilogy.”
Wilder's co-promoter Shelly Finkel added: “We're not even looking at April.
“After the huge fight on Saturday, May or June is the right time for the right fight, the natural fight, against Fury.”
DRINKS ARE ON ME
TYSON FURY'S controversial draw with Deontay Wilder disappointed many in the UK – but not this lucky punter.
Professional boxer Ted Cheeseman uploaded footage of the moment his friend won £13,000 on a £500 bet at odds of 25/1.
In the early hours of Sunday morning, Cheeseman and his friends were holed up in a pub watching the action play out in Los Angeles.
As those in the Staples Center, and around the world waited for the result to be read out, the punter held out his phone with the bet on the screen.
After a quick swig of lager the man moves closer to the screen to hear the scores.
Click here to watch his epic reaction.
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