"Anthony, I am so glad we finally heard from you on Saturday and that you want to fight me as your next opponent and you want the fight to happen in the UK," Wilder, 32, said in a statement.
"I accept that challenge and I am ready to come to the UK for my next fight. There is nothing on Team Wilder's side to prevent me from fighting you next.
"Thanks Anthony, I can't wait to meet you in the ring."
The WBC title is the only one remaining on Joshua's radar after he added the WBO championship to his WBA, IBF and IBO titles by defeating New Zealand's previously unbeaten Parker on Saturday by unanimous decision.
(Reporting by Rory Carroll; Editing by Peter Rutherford)
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