According to reports, the contest will take place on 31 March, at Cardiff's Principality Stadium.
The contract is expected to be signed and a statement will then be released by the fighters' respective promoters, Matchroom Boxing and Duco Events.
"I was bantering with David Higgins yesterday".
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", he told Sky Sports.
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"We feel otherwise. You're going to see a lot of fireworks this press conference".
Higgins also revealed that Parker remains in New Zealand while he tries to finalise a deal for the two to meet in the ring.
"This could be the first time we've seen the switch of Anthony Joshua, and I'm excited to see it".
American puncher Deontay Wilder holds the WBC title and will be the next target for whoever comes out on top when Joshua and Parker go head-to-head in the United Kingdom this spring. To do that he has to beat Parker and Wilder. The Parker fight is very close.
"But it's interesting, let's just say I've done the hard work, I've managed to keep the four belts and I've got to put everything on the line to bring them back home so it's a real tough one".