Joshua holds the IBF, WBA and IBO heavyweight titles, while Parker holds the WBO crown. American Deontay Wilder holds the WBC title.
"Eddie Hearn and I have exchanged emails," David Higgins told reporters in Auckland about talks with Joshua's promoter.
"My message was Joe would fight Joshua next and Hearn's message was that Joshua would fight Parker next.
"It's now a question of the deal and money."
Higgins said if they were able to agree to a fight it would probably be next March or April.
He added he had also been in talks with Wilder's camp. The American knocked out Bermane Stiverne in New York last Saturday and then accused Joshua of 'dodging' a showdown.
"They have put a figure on the table. It wasn't enough," Higgins said. "If they put the right amount of money on the table we will go to America to fight Wilder."
British media have reported that Joshua had indicated he would like to fight Wilder by the middle of 2018.
Parker's trainer, Kevin Barry, said he was keen to get the 25-year-old in the ring as soon as possible against the Briton.
Joshua defeated late replacement Carlos Takam in Cardiff last month and Barry said the France-based Cameroonian had exposed the Briton's weaknesses before the 10th-round stoppage.
"That Joshua fight puts everything into perspective now," Barry said. "Takam had done no sparring. He tested Joshua right to the end. We saw how gassed Joshua was.
"We all saw that Joshua could not hurt Takam. That showed me that he is very beatable. He can't knock you out with one punch. He is very robotic.
"I think the guy is there for the taking and I want us to get in front of him before someone else beats him."
If the Joshua bout did not come off then Parker could fight unbeaten Australian Lucas Browne instead, Higgins said.
"We have been in talks for a few weeks with his promoters," Higgins added. "There is no deal. We have loosely agreed terms and principles ... and it is possible if we decide it is the best option."
Barry said Parker would now enter a training camp in Auckland before heading back to Las Vegas in mid-January.
(Reporting by Greg Stutchbury; Editing by Peter Rutherford)
