Joshua added the WBA world heavyweight title to his IBF strap earlier this year, knocking out Wladimir Klitschko in front of 90,000 fans at Wembley Stadium.
The Brit then stopped his IBF mandatory Carlos Takam, a late replacement for Kubrat Pulev, at a sold-out Principality Stadium.
In the aftermath of his 20th win, promoter Eddie Hearn outlined their plans to unify the heavyweight division in 2018, a feat that will require Joshua to beat WBC king Deontay Wilder and WBO champ Parker.
The latter has been in talks with Joshua for weeks with a bitter argument over payment for each fighter stalling talks.
However, Express Sport understands that the purse split has now been agreed with Parker receiving between 30 and 35 per cent of the cash on the table.
A return to the Principality Stadium, either on March 31 or April 7, is considered the favoured location for the bout but Twickenham and Manchester venues remain in the mix.
The fight contract has been sent to Duco Events promoter David Higgins, who represents Parker, with minor details set to be finalised in the coming weeks.
The New Zealander will fly to London when he is ready to put pen to paper on the deal but says the deal is a lot closer than it was before Christmas after a “giant step forward” in negotiations.
“I’d like to pay tribute to Anthony Joshua and Eddie Hearn,” Higgins said.
“They’ve both said they want to unify. Talk is cheap but by stepping up on this occasion they’re proving that they do actually want to unify as does our side and Joseph Parker.”
Higgins also reiterated his belief that Joshua is not the supreme champion many believe him to be.
“Most of the blinded British fans think Joshua is this unbeatable messiah with no chinks in his armour when, in fact, there are many,” Higgins added.
“By the time I’ve finished my job they’re going to think Parker’s the favourite.
“I think Eddie Hearn is making a grave mistake feeding Anthony Joshua to Joseph Parker.
“The reason is simple: Joshua has a bodybuilder physique but he’s a bit predictable, robotic and he’s been flattened at least half a dozen times so his chin is suspect.
“Parker on the other hand is fast, fluid and has never, ever been dropped.”
Should the deal between Parker and Joshua fall through, Hearn’s heavyweight will make a voluntary defence of his titles on March 24 at the O2 Arena.