"We have agreed to the terms that Eddie has put out to us for a fight in the UK", Wilder was quoted by ESPN as saying.
Joshua and promoter Eddie Hearn have yet to publicly respond, though Hearn has previously criticised the Wilder team for appearing to do their business via social media, insisting "deals are made in the boardroom, not on Instagram". The fact that both men are the height of National Basketball Association stars and have devastating knockout power will make the bout, should it happen, a must see for fans - even casual fans.
The WBC champion revealed he was willing to give Joshua home advantage for the huge world heavyweight fight for all the major world titles, and co-manager Shelly Finkel confirmed that terms had been accepted.
Wilder had initially attempted to lure Joshua to the USA with a $50m offer.
"BREAKING NEWS for all you Anthony Joshua fans", Wilder wrote on Twitter.
Wilder announced today that he accepted Joshua's offer to fight in the United Kingdom next up and has also given him the option of going to the USA too of $50 million.
"We see this as a Wembley fight and we're quite limited in that respect", said Hearn.
If the Wilder fight is next up for Joshua, then the original plan of September 15 is now highly unlikely due to the lack of time involved, as well as the U.S. television clash with Canelo Alvarez's return from a drug's ban.
Starsport understands it is considerably more than the £9.3m that Hearn originally offered Wilder back in early April.
"If they keep their word, then we've got a fight", Deas said.
Deas said he's not authorized to discuss details of the potential purse and that the venue and date are up to Joshua's camp.
Hearn told Sky Sports News it is the only way to find out if the WBC titleholder Wilder is serious about fighting Joshua even though reports out of the United States on Monday suggested he had accepted a two fight deal with the 2012 Olympic champion and now holder of the WBA, WBO and IBF belts. But we need to get the contract back from Deontay Wilder - then we will know if this is bluster or if he is serious'.
Hearn has already stated that a mandatory challenge against Alexander Povetkin could be next with Wilder straight after, but the American's concession may have ripped up the plan.