Wilder, 32, accepted the terms offered by Joshua's promoters, Matchroom Sport, late on Monday evening.
Wilder's team have also left the £37m offer made to Joshua in April on the table if the WBA, WBO and IBF champion changes his mind and wants to travel to the U.S. for the fight instead.
According to Pugmire, Wilder's and Joshua's respective teams were quibbling over the location of the fight, with Wilder's camp wanting it in the USA and Joshua's side preferring the UK.
Deontay Wilder claims he has agreed to fight Anthony Joshua next in the United Kingdom, and says it is now up to the Briton's camp to finalise the deal. "Deontay wrote to Joshua and said 'Hey, I accept your offer, let's fight in the U.K'". Then they came back with a much, much, much smaller offer for Deontay, which would have been even less than Deontay would have made in the United States, and wanted him to still travel to the U.K. for the fight.
"Deontay has accepted his terms to fight in the UK", Finkel said.
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 victor.
He added: "If they change their mind, the $50 million [against 50 per cent of the event] we offered them [to come to the United States] is still on the table".
The offer also includes a rematch clause, which Joshua and his team insisted on, and Wilder is happy to accept that also. Hopefully, we'll get an answer in the next day or two. Not as good as we feel it should be, but we want the fight and we're willing to take what they offered.
Joshua, who won his first world title by stopping Charles Martin in 2016, scored his biggest victory to date the following year by rising from the canvas to knock out former champion Wladimir Klitschko in front of 90,000 spectators in London.