The heavyweight bout is expected to be confirmed within the next 24 hours and would reportedly take place on March 31 at Wales' Cardiff's Principality Stadium.
Higgins also revealed that Parker remains in New Zealand while he tries to finalise a deal for the two to meet in the ring.
Parker's promoter, Davie Higgins, is now in England and seems to have finalised a deal with Joshua's promoter, Eddie Hearn. They truly believe they knock Anthony Joshua out.
Kohli confident India will bounce back against the Proteas
We are a team that looks to fight back and get right back into the game, we are not a team that gives opposition easy victories. You still need to have a solid technique and go out there and be ready to face spells, which are going to be hard .
"We feel otherwise. You're going to see a lot of fireworks this press conference".
"This could be the first time we've seen the switch of Anthony Joshua, and I'm excited to see it".
It is thought that the process has started by Fury and his team about applying for his boxing license and the Ring Magazine heavyweight champion this week stated he'll fight again this April. To do that he has to beat Parker and Wilder. The Parker fight is very close.
Talks have been quite amicable, meaning Parker was happy enough to make the long trip to the United Kingdom to take part in a press conference to confirm the fight early next week.