Fury posted a video on social media this morning accepting the bout.
However, the Gyspy King will only accept the deal if the pair fight for the diamond belt instead of the interim title.
Fury feels he should already hold the WBC heavyweight belt following his controversial draw with Deontay Wilder last year.
“I just want to send a special message to the WBC who have supported me in my comeback and have been very fair with me,” Fury said.
“I recently saw your quote about fighting for the interim title against Dillian Whyte, and I thank you for your interest.
“But because I beat Deontay Wilder, though I did’t get the decision I still won and the world knows.
“I propose that we make me and Dillian Whyte for the WBC diamond belt, not the interim belt.
“I will take care of Dillian Whyte and I will knock him out within six rounds.
“I accept the challenge. Thank you very much. I will fight Dillian Whyte every day of the week and twice on a Sunday.
“Let’s get it on!”
Fury must first focus on beating Tom Schwartz on June 15 before any potential negotiations with Whyte.
Dillian Whyte has since responded with a message of his own.
“Let’s do it then, Tyson Fury,” he said.
“I’d fight you anytime, anywhere, seven days a week and five times on Sunday I look forward to putting you to sleep.
“Let’s go baby.”