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Joshua (31) holds the IBF, WBO, WBA and IBO titles while Fury (32) is the WBC belt holder.
"August the seventh, August the 14th. I think it's a very bad secret that the fight is happening in Saudi Arabia," Hearn, head of Matchroom Sport and Joshua's promoter, told Sky Sports television.
"It's the same people we did the deal with for Andy Ruiz, that event was spectacular. As partners they were fantastic as well," he added.
Joshua won back his belts from Andy Ruiz Jr in Diriyah, on the outskirts of the Saudi capital Riyadh, in December 2019. He had lost the belts to the Mexican-American in an earlier shock defeat.
Hearn said Fury's lawyers were going through the fine details but an announcement could come this week.
"We're very comfortable. Anthony's comfortable, he knows those people. They delivered on every one of their promises last time, we're ready to go," said Hearn.
He said it made more sense to go for the 14th in terms of the global spectacle, with the Tokyo Olympics finishing on August 8, but hoped to nail down the date and venue in the next few days.
Hearn said in March that Fury and Joshua had agreed a two-fight deal to unify the title, although Fury's co-promoters subsequently cast doubt on that.
Fury has not fought for more than a year since his knockout win against American Deontay Wilder. Joshua's last fight was against Bulgarian Kubrat Pulev in December.
Joshua and Fury have both been active on social media recently, trading barbs.
Fury said in a podcast with Hearn last week that he would expose Joshua in the ring as a fake and cut him down "like a hot knife through cheese".
He said the real difference between them was that Joshua was a businessman, while he was a 'Spartan'.
Joshua took to Twitter on Tuesday to say: "Spartan" I’m tired. My fans are tired. Your fans are tired. We're all tired of the talk. Less talk. More action!"
Fury replied by saying Joshua was "no talk no action".
Hearn said he would be talking to Fury's people later on Tuesday.
"We're in a great place, we're very comfortable with the deal now and we want to move forward and we want to get it done," he said. "I want to be able to stand here and not say 'give me a couple of days or next week'.
"We just have to get everybody comfortable to shake hands and say 'we're in this as a team, we're in this for boxing. Let's move forward and make this fight'."