The WBC title holder, who has 40 wins from 40 bouts, holds the only heavyweight belt not owned by Joshua, who added the WBO gong to his IBF and WBA haul against Joseph Parker on Saturday night.
The American boxer postponed an international media call scheduled for Tuesday, instead releasing a statement declaring his intention to find a resolution that will see him go toe to toe with Joshua in the UK.
"First of all, I want to congratulate Anthony Joshua on his win last Saturday night," Wilder's statement read.
"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.
"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.
"They are also ready to meet with your team immediately. Let us know when - the sooner the better. Thanks Anthony, I can't wait to meet you in the ring."
Neither Joshua nor Wilder has lost a professional boxing fight.