Joshua's promoter Eddie Hearn has told Sky Sports News that negotiations are down to "single digit" percentages.
Parker and promoter David Higgins laid out their final negotiating position in Auckland last week, asking for 35% of the bout's net profits.
"We feel like there is a deal that can be in place with Parker and his team within the next week or so," Hearn told Sky Sports News.
It's been tipped that the two will fight at London's O2 Arena in March. Parker is the current WBO heavyweight champion while Joshua holds the WBA and the IBF crowns.
"There's a good chance [Parker's promoter] David Higgins will fly over in the next week to try and close the deal," he exclusively added.
"Sometimes when you're this close you need to get face-to-face and eye-to-eye. Work on it, keep talking, no one goes home until you make the deal.
"There are two perfect fights for AJ - Parker and Deontay Wilder. The Wilder fight is unrealistic because of what they want. The Parker fight is a lot closer."
