The heavyweights have agreed to two fights, with the first expected to take place in the summer at a currently undisclosed venue.
Joshua, 31, will hit up the EIS in Sheffield with his trainer Rob McCracken to prepare for the fight.
Hearn told the press: "We can't wait. AJ has started his training camp and Fury has been training very hard. They all want the fight.
'They are going to be presented with a few proposals on where to do the fight and everyone has to agree on where to do it.
"There is no other fight out there that comes close for either guy. July is the date. Really I think end of next week it will be done.
"Talks have progressed extremely well and we're closing in on securing the venue and there will be an announcement in due course.
"We're at the stage where it's probably best not to say much more than that."
Fury, meanwhile, has been working alongside Joseph Parker, who fought Joshua three years ago.