All four teams will take part.
New Zealand will host Australia in two Bledisloe Cup tests before the Rugby Championship.
The tournament was moved from its traditional August-September window due to the ongoing effects of the global COVID-19 pandemic.
The tournament will comprise a six-week, 12-match, "home and away" tournament involving Argentina, Australia, New Zealand and South Africa.
NZR CEO Mark Robinson said: "We're obviously disappointed at the decision to not have New Zealand host the tournament, but we understand and accept it. Now that the decision has been made, we're looking forward to seeing the All Blacks again take on the best of the Southern Hemisphere.
"We worked incredibly hard behind the scenes with a whole range of stakeholders, including SANZAAR and the New Zealand Government, to ensure we were ready and able to host the Championship and we felt we were. We'd like to thank everyone involved for working so hard on the planning to have the tournament here."
"Those two matches (in New Zealand) will be massive for our fans and the All Blacks. We know that the Bledisloe Cup is the pinnacle of trans-Tasman rivalry and there will be huge anticipation ahead of those matches," Robinson said.
New Zealand Rugby is yet to announce dates or venues for those matches.
SANZAAR chief executive Andy Marinos explained their decision to name Australia as hosts of the Rugby Championship.
"SANZAAR ultimately determined that based on government-required quarantine protocols [for entry and training prior to the tournament] and commercial underwriting, the Rugby Australia submission was the most desirable and workable in terms of tournament logistics for the essential pre-tournament preparation period and the six-week tournament itself."
Marinos stated that South Africa's participation remained to be confirmed as it was dependent on several factors outside the control of SANZAAR, "The progress and impact of the pandemic has varied from country-to-country and international sporting competition is currently suspended in South Africa," he said.
"The Springboks' participation will be dependent on the relaxation of that suspension as well as overcoming a number of other logistical challenges including the opening of international air borders. South Africa is only expected to return to competitive play next month [October], leaving a relatively short time to prepare."
SANZAAR is now currently working through the refinement of the detailed planning with Rugby Australia and we hope to announce match venues, match dates and kick-off times in the very near future."
Rugby Australia has announced that one round will be held in Queensland with the rest of the games in New South Wales.
"We are cognisant of the fact that the COVID pandemic environment continues to change and throw up obstacles but we are confident that our plans are robust and we will naturally build in contingency measures to suit any further changes," added Marinos.