The American, 32, holds the WBC title and wants a unification bout with Joshua, the WBA, IBF and WBO champion, BBC Sport reported.
He tweeted, "The $50m offer for him to fight me next in the US is still available. Today I even agreed to their offer to fight Joshua next in the UK."
Joshua's promoter Eddie Hearn told Sky Sports he will send the contracts but wonders if Wilder's claim is "bluster".
He added, "It's all come as a bit of a shock that Deontay's finally come back to us on the offer and seems to accept the terms.
"One thing I can tell you is that by the end of this week Deontay Wilder will have a contract in front of him and we'll see if he's real."
Hearn has previously criticized the Wilder team for appearing to do their business via social media, insisting "deals are made in the boardroom, not on Instagram".
"If he prefers the fight in the UK, the ball is in their court. It's up to them to choose," Wilder added.
Wilder's manager Shelly Finkel told ESPN they had officially accepted terms to a fight in the UK.
He said, "Deontay sent an email to Joshua [Sunday] night, and I sent one [Monday] to Barry Hearn and Eddie telling them that we officially accept the offer to fight under the terms they gave us and to send us the contract."
Should Joshua, 28, fight the American next, the winner would become the first heavyweight to hold all four recognized world titles at once.
Russian Alexander Povetkin is the WBA's mandatory challenger and is close to a deal to fight Joshua, although Joshua could face Wilder first if Povetkin is able to fight the winner.