But promoter Eddie Hearn has warned a deal must be agreed quickly if the American is to be Joshua's next opponent.
Joshua successfully defended his International Boxing Federation, World Boxing Association and World Boxing Organisation belts with brutal a seventh-round stoppage of Alexander Povetkin at London's Wembley Stadium on Saturday.
He is already booked in to box again at Wembley on April 13, but the question of an opponent has still to be resolved.
Joshua has repeatedly stressed he wants to fight fellow undefeated champion Wilder, who holds the World Boxing Council version of the heavyweight title, next.
"We have to fight, it would be silly not to" Joshua told reporters after inflicting the first stoppage-loss of Povetkin's professional career.
Wilder, however, is next due to face Britain's former world heavyweight champion Tyson Fury on December 1.
Talks between the Joshua and Wilder camps have stalled and Hearn is worried a fight that will be worth far more than the "peanuts" of an initially offered $50 million could be derailed.
If Fury beats Wilder, the American's re-match clause would likely be activated, delaying both boxers from facing Joshua, the London 2012 Olympic champion who now has a professional record of 22 wins from 22 fights, with 21 knockouts.
"Being British, we'd like Fury to win, but for April, Wilder must win if that's going to happen," said Hearn.
"We're not willing to wait until December to see. A deal must be done in advance of that, subject to him winning.
"But now, after 80,000 (the estimated crowd at Wembley) and the worldwide exposure and after the finish (against Povetkin) that offer (made previously, by the Wilder camp) will look like absolute peanuts when this fight is made."
"That is the biggest fight in boxing; Wilder-Fury is a really good fight to see who's the second best heavyweight in the world. Joshua-Wladimir Klitschko was the biggest fight in British boxing history, and Joshua-Wilder would eclipse that. We want that now."
Hearn insisted: "We're not waiting for time-wasters; we understand if they lose we'll have to find another opponent, but if you win, we're not waiting until December.
"These aren't negotiations that will take 24 hours, and Joshua's career is not being slowed down. If they don't want to do that, we'll fight someone else."
Fury has previously said on social media that Joshua will never fight him but Hearn witheringly added: "Tyson Fury is the least entertaining fighter I've ever seen.
"He's never been in a good fight, apart from against Steve Cunningham (in 2013), when he got knocked down."
Meanwhile the 28-year-old Joshua said he too had no intention of being messed about by Wilder.
"If Wilder's not serious, there's other people out there; when he's ready, we're ready."
Joshua added: "Good luck to them both (Wilder and Fury) – boxing needs it.
"I've had the burden of the heavyweight division on my back for some years, because it was all about me fighting Wilder, Fury, Klitschko, Dillian (Whyte), Povetkin. That's all they were interested in – me fighting them all.
"So I'm happy those two are fighting. April 13 is booked, so whichever heavyweight is serious, we can look at making a deal. I have no interest in who wins; I'm not fussed."
Joshua was rocked by Povetkin, the 2004 Olympic champion, in the first round. But he insisted the 39-year-old Russian had not broken his nose.
"These guys the last thing they lose is their power, but it was a good way to wake up," Joshua said.
As for what lay behind his own public appeal, Joshua added: "The appeal is losing – who is going to be the man to beat me? Sometimes you have to go in there and really earn your money.
"Povetkin was a tough challenger for sure, but I knew how to break him down.
"I wasn't looking for the knockout but the instinct told he was hurt. I knew how to tidy up and I knew it was time to get out of there."AFP