Hearn held another meeting with Parker's promoter David Higgins on Wednesday, as they continued to discuss the remaining details of a deal for Joshua to put his WBA "super" and IBF heavyweight titles at stake against the WBO champion.
According to reports, the contest will take place on 31 March, at Cardiff's Principality Stadium.
The fighters' respective promotions, Matchroom Boxing and Duco Events, have been locked in negotiations since the end of previous year, with the purse split the largest obstacle that needed to be hurdled.
Parker's representative, David Higgins, has been in the United Kingdom since Monday to conclude negotiations, and a news conference is expected next week.
Boxing champion Anthony Joshua met legendary actor Tom Hanks as they graced their presence at "The Graham Norton Show" - an English celebrity TV chat show. It's not even in question for them.
Parker's promoter, Davie Higgins, is now in England and seems to have finalised a deal with Joshua's promoter, Eddie Hearn.
"We feel otherwise. You're going to see a lot of fireworks this press conference".
Meanwhile, Joshua is in talks for a unification bout against New Zealand's Joseph Parker.
"Deontay Wilder, he's got a big mouth", Joshua replied when asked who he'd face after Parker.
"We need these belts, there is no secret, and they know that as well", he said. The Parker fight is very close.
"We might get Parker on a plane this weekend and all being well, hopefully we can announce the fight next week".