According to Times Media Group, the 63-year-old made it clear to the country's FA that he would accept the offer if they allow him choose his backroom staff.
It is believed that Baxter wants his son, Lee, to become Bafana's goalkeeper coach.
The two worked together at SuperSport last season. Lee also worked with Baxter as a goalkeeper coach at AIK and Genclerbirligi.
Joshua Smith, who's currently part of Baxter's staff at Matsatsantsa as a fitness trainer, is also expected to follow the former Kaizer Chiefs mentor to Bafana.
The report suggets that Safa were initially reluctant to accept Baxter's terms, but they eventually gave in.
Baxter will reportedly be unveiled either on Monday or Tuesday, but he's expected to honour his contract with SuperSport until the end of the season.
Safa is under immense pressure to appoint Shakes Mashaba's successor ahead of this year's Africa Cup of Nations (Afcon) qualifying campaign which gets underway in June.