The unbeaten Mancunian made the claim after speaking to Haye’s manager Adam Booth at James DeGale's successful WBC super-middleweight silver title defence against Stjepan Bozic in Bluewater.
Speaking on Channel 5, the 25-year-old Fury said: "I was minding my own business over there and I bumped into Adam Booth, so we got him on camera and we asked him: 'Why is the David Haye fight not being done,' and he said: 'Do you want the fight?' so I said: 'Two million per cent. The ball is in your court so call me tomorrow morning.'
"He said: 'I cant see why this fight cannot be done' - so there it is from the horse's mouth. Adam Booth says he wants this fight. The fight is 99 per cent baby!"
Talks had been reportedly ongoing for some time, but Fury's comments imply any contact was only made recently.
Fury later told SportsMail that he would love to fight Haye in Manchester, but would be happy with London.
"I'd love it to happen in Manchester but we might flip a coin to see where it happens.
"I can beat David Haye because I'm the greatest fighter that's ever lived; that's ever stepped out of Britain.
"My plan is to knock out Haye, then Klitschko, Klitschko. And then vacate a title for my younger cousin Hughie Fury to beat Mike Tyson's record to become youngest ever heavyweight champion," Fury added.
An injured Fury recently pulled out of an IBF eliminator clash with Kubrat Pulev, which would have given the winner a mandatory challenge against champion Wladimir Klitschko.
Haye, 32, is a former WBA world champion but failed in his bid to take Wladimir's three belts in 2011. He then retired, although reversed his decision to fight fellow Londoner Dereck Chisora in 2012.