Fury, the self-styled “Gypsy King”, took to social media after Pulev was knocked out, saying he would knock out his fellow British heavyweight “inside three rounds”.
An agreement in principle for Joshua, who retained his WBA, WBO, and IBF belts, and WBC champion Fury for two meetings in 2021 has been in place since earlier this year.
Asked how long it would take to complete a deal, Hearn replied: “A couple of days.
“There is no reason we cannot complete the deal and then move on and solve the other problems.
“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.”
There are some potential stumbling blocks, including if Fury is forced to fight Deontay Wilder for the third time and Joshua could be told by the WBO to face mandatory challenger Oleksandr Usyk.
But Hearn insisted: “The only hitch I can think of is Wilder. If the court orders the Fury rematch, and it doesn’t sound as though they will, then it is out of our hands.
“That is the only way it falls through is if something stops their side. That isn’t me saying they don’t want it — they do. But if the court orders it then we will have to fight Usyk.”
Matchroom promoter Hearn disputed Fury’s gung-ho assessment of his chances, saying Joshua would knock him out.
“AJ got hit once maybe with a right hand (against Pulev), that was about it,” he said. “Fury and Pulev probably punch as hard as each other, Fury is not a devastating puncher, but he is a lot better than Pulev.
“It is an undisputed fight, it should be the toughest fight out there, but I’ve always fancied him to beat Tyson Fury, always.”
Fury has revealed he has asked his lawyers to help remove him from the BBC’s sports personality of the year shortlist.
The heavyweight champion said he asked “politely” when he posted an Instagram message earlier this month.
But he has now taken further steps to be removed from the list of contenders and instructed his lawyers to send a letter to the BBC.
Fury’s previous appearance on the shortlist in 2015 caused controversy, with more than 100,000 people signing a petition calling for the boxer’s removal over a series of outspoken comments on homophobia and abortion.
But in 2018 Fury’s omission from the final list, following his draw against Wilder, drew criticism from the boxer’s camp.
This year’s shortlist also includes Formula One world champion, world snooker champion Ronnie O’Sullivan and Liverpool captain Jordan Henderson.