Rohit and Kohli were part of the India team who lost to Australia in the final of the 50-overs home World Cup earlier this month.
In Rohit's absence, Suryakumar Yadav will lead the side in the three-match T20 series beginning in Durban on December 10th.
Wicketkeeper-batsman KL Rahul will take charge for the three one-day internationals that follow before Rohit and Kohli return for the two-Test series.
"Rohit Sharma and Virat Kohli had requested the board for a break from the white-ball leg of the tour," the Board of Control for Cricket in India said in a statement.
India dropped middle-order batter Ajinkya Rahane while Cheteshwar Pujara could not force his way back in, with both seemingly at the end of their international careers.
Seamer Mohammed Shami was picked for the Tests in Centurion and Cape Town subject to him passing a fitness test.
India's Test squad:
Rohit Sharma (captain), Shubman Gill, Yashasvi Jaiswal, Virat Kohli, Shreyas Iyer, Ruturaj Gaikwad, Ishan Kishan, KL Rahul, Ravichandran Ashwin, Ravindra Jadeja, Shardul Thakur, Mohammed Siraj, Mukesh Kumar, Mohammed Shami, Jasprit Bumrah, Prasidh Krishna'
India's T20 squad:
Yashasvi Jaiswal, Shubman Gill, Ruturaj Gaikwad, Tilak Varma, Suryakumar Yadav (captain), Rinku Singh, Shreyas Iyer, Ishan Kishan, Jitesh Sharma, Ravindra Jadeja, Washington Sundar, Ravi Bishnoi, Kuldeep Yadav, Arshdeep Singh, Mohammed Siraj, Mukesh Kumar, Deepak Chahar
India's ODI squad:
Ruturaj Gaikwad, Sai Sudharsan, Tilak Varma, Rajat Patidar, Rinku Singh, Shreyas Iyer, KL Rahul (captain), Sanju Samson, Axar Patel, Washington Sundar, Kuldeep Yadav, Yuzvendra Chahal, Mukesh Kumar, Avesh Khan, Arshdeep Singh, Deepak Chahar
