A coronavirus-induced ban on international sport by the South African government was lifted recently and the national football team hosted Namibia this week.
But South African Rugby officials are seeking further clarification from the sports ministry as to how the regulations governing participation would affect the Springboks.
According to the statement, a decision will be made "early next week" on whether South Africa will defend a title won last year in a single-round tournament because of the Rugby World Cup.
Competitive rugby resumed in South Africa only this weekend after a six-month suspension triggered by the pandemic and there is concern about the readiness of players for Test rugby.
Springboks coach Jacques Nienaber has said he would not favour competing unless the chosen players had been involved in a minimum of six matches.
Given the much later return to rugby than New Zealand and Australia as South Africa was more affected by the virus, that target cannot be met.
Officials also have to consider the financial impact of not competing against host Australia, New Zealand and Argentina in the 31 October to 5 December tournament.
SA Rugby slashed 1.2 billion rand ($100.7 million) off its 2020 budget because of the coronavirus and reportedly could pocket 300 million rand ($25.2 million) by playing in Australia.
If the Springboks opt to participate, they will begin their title defence against Argentina in Brisbane on 7 November.
The Rugby Championship was created in 2012 when Argentina joined a competition previously called the Tri-Nations.
It is normally staged between August and October with each country playing the other three at home and away.