Brown has previously said he has twice rejected the advances of NZR, who wanted him to join the Ian Foster and Robertson coaching staffs.
Last month, Erasmus re-signed to stay on as Springboks coach until the end of the 2031 Rugby World Cup.
When SA Rugby made the announcement, it added that: “An announcement on the contract negotiations with other members of the Springbok coaching team will be made in due course.”
There has been lots of public support for Joseph to take the reins as All Blacks coach, including from former player and Sky Sport pundit Jeff Wilson.
“For me, the logical choice is Jamie Joseph,” Wilson told Newstalk ZB. “He went head‑to‑head with Scott Robertson last time and, to my understanding, came close. Jamie Joseph is the obvious man.”
Joseph is set to coach the Highlanders in the upcoming Super Rugby Pacific season, while other names linked to the job include Dave Rennie, Robbie Deans and Sir Steve Hansen.
Former Chiefs coach Clayton McMillan has already ruled himself out of joining the All Blacks coaching ranks, insisting he’s committed to his current club, Munster.
The rugby world was left stunned on Thursday when Scott Robertson was relieved of his duties as All Blacks coach, two years into a four-year contract.