SAFA announced it would not appeal the order that annulled South Africa's 2-1 win over Senegal last November, but said that its "rights to sue for damages remain open."
That raised the possibility that SAFA might seek compensation from FIFA, which it believes should take responsibility for the referee, who was appointed by FIFA's referees committee despite some suspicious performances.
"Why didn't FIFA do an integrity check on the referee since they have these mechanisms?" SAFA senior legal counsel Norman Arendse said.
FIFA's decision on the South Africa v Senegal qualifier was the first time the world body ordered a game to be played again because of match-fixing.
FIFA banned Ghanaian referee Joseph Lamptey for life after he awarded South Africa a penalty for a non-existent handball against a Senegal defender. South Africa scored from the penalty and went on to win the qualifier 2-1.
FIFA found neither team was involved in the fixing, and the referee was likely manipulating the match for illegal betting rings.
Lamptey had previously been under scrutiny and was suspended by the Confederation of African Football for allowing a crucial goal from a clear handball to stand in an African Champions League semifinal match in 2010.
The fact that he was still appointed by FIFA's referees committee for the World Cup qualifier is one of two major gripes for South Africa. The other is that SAFA was never invited to make any representation in the process by FIFA, which ultimately ended up in a decision that adversely affected South Africa.
South Africa now has little chance of reaching the World Cup in Russia next year.
FIFA's unprecedented decision to order a replay because of fixing must be confirmed by its organizing committee for competitions at a meeting on Thursday. It should be rubber-stamped, with no opposition now from SAFA.
"SAFA has decided on ethical and moral grounds that if this match was manipulated, we should replay it," the South African federation said.
The game will be replayed in November, likely after all the other qualifiers in the group are completed.
While FIFA's decision badly hampers South Africa's World Cup hopes, its gives Senegal a second chance. Currently third in the group, a victory in the replay for Senegal could send it to the World Cup at the expense of the other three teams.