Haye, who lost his WBA belt in July 2011 to Ukrainian Vladimir Klitschko, was sparring with Croatian Filip Hrgovic on Friday when he suffered the wound that needed six stitches.
Promoters Hayemaker Boxing said in a statement that the stitches would have to remain in place for between five and seven days. No date was set for any rescheduled fight.
"Even when everything seems to be right on track, sometimes life throws you a curveball," said the 32-year-old Haye. "I will somehow have to try and make the best of this horrible situation. But, right now, I'm in bits about what has happened."
Haye's previous planned fight, against Germany's Manuel Charr, was cancelled in June when the Briton suffered a hand injury. He has not fought since July 2012, when he beat compatriot Dereck Chisora.
Fury is a former British and Commonwealth champion.
(Reporting by Alan Baldwin, editing by Ed Osmond)
