Hearn confirmed on Tuesday that further talks will take place with Parker's representatives in the coming days in the hope of getting a bout agreed.
Unbeaten duo Joshua and Parker – the WBO champion – are expected to go toe-to-toe in March or April, although negotiations have thus far been protracted.
The split of the purse has been the main reported problem for getting a deal signed, but a compromise is seemingly close to being found.
"[Parker's promoter] David Higgins landed this morning and we are meeting this afternoon to plough the agreement and solve any outstanding issues," Hearn told Sky Sports.
"If we can do that then it's likely Joseph Parker will jump on a plane towards the end of the week.
— Anthony Joshua (@anthonyfjoshua) January 8, 2018
Higgins was equally as optimistic as talks are set to resume, but insisted there were still some issues to "iron out".
He said: "There's no deal yet, but we have a draft contract going over and back, and some of the big tricky things have been agreed which is promising.
"I think it's also fair to say that Eddie Hearn and I both have a will to try and make a deal, which counts for something.
"Sometimes people don't want to make a deal and it's obvious, and they use excuses, but from my experience this is a case where it appears both sides actually want to try and make a deal. That counts for a lot.
"We have both set aside quite a bit of time to iron out the final details. Again, nothing until the contracts are signed."
Photo Getty Images. Caption: Anthony Joshua and Joseph Parker