Reigning WBA super and IBF title holder Joshua has already outlined his plans for 2018, which involves facing both Parker and Deontay Wilder en route to possibly becoming undisputed heavyweight champion.
Hearn is now more confident of getting a date pencilled in for next spring, however, with an agreement potentially just days away.
"We're getting there. As AJ says, he wants the belts, he wants to be the undisputed king of the division. That's the aim, and to do that he has to win two belts," he told Sky Sports News.
"We're looking at March 24, March 31 and April 7 as potential dates for his next fight, with various different venues in London and Wales, even other venues and cities around Europe as well.
"We want that unification fight, we're close to a deal with Parker - it's not done yet but we hope over the Christmas period, the turkey will get put to one side and we'll work to get that done.
"Hopefully we'll be in position to announce that early in 2018. AJ is that Willy Wonka fight - the golden ticket, the one everybody wants. A lot of people are talking 'I want this, I want that' but if you really want it be realistic and come get it."
Joshua has won all 20 of his professional bouts by knockout, while Parker is undefeated in 24 fights.