Joshua was quick to call out Wilder after defeating Joseph Parker on Saturday to add the WBO title to his own IBF and WBA belts, The Independent reported.
Wilder's co-manager Shelly Finkel has responded to the challenge, saying the American fighter's team are prepared to begin immediate negotiations over a unification contest.
"We are really glad that Anthony Joshua said for the first time, 'I will fight Deontay next...' and we believe he is a man of his word. Deontay accepts the challenge," Finkel said.
"We want to make it public that Deontay wants the fight, and we are prepared to come over, or meet here and get the deal done.
"Deontay is ready to sign and come to the UK to fight this summer."
Only WBC champion Wilder, who is undefeated in 40 contests, stands in the way of the Briton unifying all the division's belts for the first time in history.
Finkel, who co-manages Wilder with Al Haymon, said he will contact Joshua's promoter Eddie Hearn this week.
Hearn said on Sunday morning that there was a small window of opportunity before Joshua must defend three other belts against mandatory challengers.
Finkel, who has the authority to get the contest signed from the Wilder side, responded, "We're ready. Let's do it. Unification fights always trump mandatories. A unification between Wilder and Joshua would push the mandatories back.
"Assuming it will be Wembley Stadium in the summer, we are ready, and if they are ready as they say they are, we want to get it done. We want the fight, they want the fight, the public wants the fight.
"There is nothing to stop it going ahead."
Finkel confirmed that talks had taken place with both Eddie Hearn and Barry Hearn last year during which the prospect of two fights, one in the UK and one in the US, were discussed.
"That was in November, we were told they would get back with a P and L (statement), with the numbers, but we never heard back from them again," Finkel added.