Joshua’s knockout of Kubrat Pulev on Saturday has set up the prospect of a clash between the British rivals.
Negotiations start on Monday and it could be the first fight in history for all four heavyweight titles.
“There is no reason we cannot complete the deal and then move on and solve the other problems,” Hearn said.
WBC world heavyweight champion Fury has been installed as an odds-on favourite to beat IBF, WBA and WBO belt holder Joshua if they fight next.
Asked how long realistically it would take to complete the deal, Hearn replied: “A couple of days.
“We need to write to the governing bodies and say yes we have a deal and we will now talk to the various sites and confirm the dates shortly for what I suppose will be the end of May.”
“I want the fight next. I can’t wait to knock him out,” Fury said in a video posted on social media on Saturday night.
Source: BBC Sports