The two Brits are finally set to meet this summer in a hugely-anticipated bout having agreed terms over a two-fight deal earlier this year.
When and where the fight will take place remain up in the air, however, with a number of options for both being considered.
However, Matchroom chief Hearn has revealed that a choice of location is close after a meeting with Fury in Las Vegas.
"Both sides have approved the site offer that they want to go with, and now we're just finalising the site deal and we're in a great place,” he told Behind The Gloves.
"I saw Tyson yesterday. I just messaged him and said, 'I'm in Vegas, I'm in your hotel if you fancy a cuppa'. We just had five or 10 minutes together and it was good, because I don't represent Tyson Fury, so I don't know what is in his mind. What he basically cemented in my mind was, this is the only fight he wants, and I know that's the same with AJ.
"It just good to say, right we've got the offer, we all approve. We're moving forward now. We're good. It has been agreed by both sides, the option that we will take, and you'll get the date very soon and the official announcement."
Saudi Arabia appears to be the frontrunner to host but there remains interest from a number of other venues including America, Qatar and Dubai.
There has been some speculation that the clash could be moved back until later in the year, but Hearn insists the plan remains for the two to meet as soon as possible.
“One hundred per cent,” he added. “That’s the thing. Everybody wants this fight as soon as possible.
“There is no option for this fight to go at the back end of the year, and I’ve made that clear. The hardest thing about securing the site is in this world we live in today, most people would like to kick it to November, December. That’s not available. It was never available.
“What was available was a summer fight for the undisputed world heavyweight championship. That is what you’re going to get. Both guys want two fights this year. One will be in the summer and one will be in December.”