The World Boxing Council last week ordered they fight in a final eliminator to challenge WBC heavyweight champion Deontay Wilder, who Fury controversially drew with in December.
Fury will fight for the first time in Las Vegas against Tom Schwarz on June 1, a month before Whyte fights Oscar Rivas in London.
Speaking in a video on social media, the 30-year-old Fury said: “I accept the challenge. I’ll fight Dillian Whyte every day of the week.”
Whyte responded by writing: “Let’s do it then. I’d fight you anytime, anywhere.”
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The 31-year-old Whyte also remains in contention for a rematch with Anthony Joshua, his long-term rival.