Etzebeth's father dies four days before he is due to lead Springboks
Lock Etzebeth is standing in as skipper because flanker Siya Kolisi is injured and will not be available to play until August, a month before the Rugby World Cup kicks off in France.
An SA Rugby official told AFP that "Springbok management will leave it up to Eben whether he wants to return home immediately or lead the team on Saturday.
"Our understanding is that Eben is very emotional at the moment as he was extremely close to his father," he added.
Should Etzebeth decide to play for the World Cup title-holders it will be his first match since suffering a shoulder injury three months ago.
The Rugby Championship has been reduced to three matchdays this season because of the World Cup, and the second series Test in Auckland is considered a probable title decider.
Both teams were impressive winners last weekend with South Africa hammering Australia 43-12 in Pretoria and New Zealand trouncing Argentina 41-12 in Mendoza.
An Etzebeth family member confirmed that Harry Etzebeth died on Tuesday night, but did not give further details. He had been suffering from cancer.