WBC title holder Wilder took to Twitter on Monday evening to say that he will travel to the UK to take part in a heavyweight unification bout with undefeated Joshua, who currently holds WBA Super, WBO and IBF belts.
Wilder said: "Breaking news for all you Anthony Joshua fans... 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. If he prefers the fight in the UK, the ball is in their court. It's up to them to choose."
Shelly Finkel, Wilder's co-manager, told ESPN: "We have agreed to the terms that Eddie [Hearn] has put out to us for a fight in the UK.
"Deontay sent an email to Joshua on Sunday night and I sent one today 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."
Wilder's most recent fight saw him defeat Luis Ortiz in March to retain his WBC title.