They were scheduled to fight on July 9 at Manchester Arena in Manchester, England, Fury's hometown.
Fury, sitting in a parked car, announced that the fight was being postponed in a video on his Instagram account.
"Hey guys, I got a little bit of bad news for you all," Fury said. "About 10 days ago, I was running and I went over my ankle, sprained my ankle. I've been to the hospital and had X-rays and MRI scans and all that stuff. They say it's not broken but it's badly sprained. Keep off it for 6-7 weeks. I tried to still train a bit. I had a second opinion a week later and I got the same advice. So the fight will be postponed.
"Sorry to all my fans to let you all down, but injuries do happen. It's one of them things. My apologies to everybody. The fight will be on as soon as possible. That's all I know at this moment. But the fight is still going ahead as soon as the ankle gets better. We will reschedule."
Fury, who was in prime condition at his training camp in Holland two weeks ago, told his supporters: ‘I got a little bad news for you all. Almost 10 days ago I was running in the Lake District and I went over and sprained my ankle. Been to hospital. Had X-rays and MRI scans.
‘Not broken but I was told to keep off it for six to seven weeks. Tried to train a bit but a second opinion was the same. So sorry to let all my fans down and apologies to everyone who’s booked the date.’
The postponement will not be helpful to Klitschko, given the strain of perfecting preparations at the age of 40.
But Fury’s promoter Mick Hennessy says: ‘There is no way we could let Tyson fight a man like Klitschko when anything less than 100 per cent fit. And we will rearrange the fight.’
Klitschko’s manager, Bernd Boente, says: ‘This is unfortunate. Wladimir was in great form. But we will be ready to go to Manchester on the new date.’
Earlier in the day, Fury responded angrily to Klitschko likening him to Adolf Hitler. Klitschko made that stark comparison after reading Fury’s views on Jewish people.