Peter Fury tweeted the news on Friday, and a further announcement is expected after 12:00 BST.
Manchester-born Fury, 27, defeated 40-year-old Ukrainian Klitschko on points in Dusseldorf last November to secure the WBA, IBF and WBO titles.
Victory meant he became Britain's fifth bona fide heavyweight world champion.
Fury was stripped of his IBF belt two weeks after winning it as his rematch with Klitschko meant he would not fight the body's mandatory challenger, Vyacheslav Glazkov.
That title is at stake on Saturday, when British Olympic gold medallist Anthony Joshua fights defending champion Charles Martin at the O2 Arena in London.
Fury's fight with Klitschko will take place on the same weekend as the Wimbledon singles finals, the British Grand Prix and the Euro 2016 final.
Tyson Fury's rematch against Wladimir Klitschko will take place in Manchester on 9 July, according to the British fighter's trainer and uncle.