Reports suggest the fight will take place on March 31 at Cardiff's Principality Stadium, which staged Joshua's most recent victory against the French challenger Carlos Takam.
Joshua, 28, holds the International Boxing Federation (IBF) and World Boxing Association (WBA) belts, while Parker, 25, who will arrive in London this week, is the World Boxing Organization (WBO) champion.
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However progress was made shortly before the new year, with Parker's co-promoter David Higgins revealing that his boxer had accepted between 30 to 35 per cent of the purse.
The fighters' respective promotions, Matchroom Boxing and Duco Events, have been locked in negotiations since the end of a year ago, with the purse split the largest obstacle that needed to be hurdled.
"My man David Higgins - he's a bit nuts, David Higgins, but he's a good guy", Hearn told IFL.TV. It's not even in question for them.
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Tyson Fury's new trainer Ben Davison has warned Anthony Joshua he could find himself knocked "clean out" by his charge's "underrated" power.
"Let's see if you're real".
The Duco Events boss says Parker has a mental edge over Joshua.
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"We have a hypothesis that Joshua is like the English football team. It's a great fight between two undefeated world heavyweight champions and we're close now", Hearn said. "We are ready to talk and negotiate a fight with Fury".
But he is back in the gym, has lost almost four stone on a new diet and is targeting an April bout to ensure he is not stripped of his The Ring magazine world heavyweight belt - as he was the WBA, IBF and IBO titles. But Parker has won all 24 of his professional fights.
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