The 25-year-old will be further assessed in six weeks by the medical team to determine his return-to-play timeline.
Jansen was not included in the national squad for the recent two-match Test series in Zimbabwe that the Proteas won 2-0.
The Proteas take on Zimbabwe and New Zealand in a T20I tri-series in Harare, starting on Monday, with the African teams set to battle in the first clash.
T20I tri-series fixtures:
Monday, 14 July –
Zimbabwe vs South Africa – Harare Sports Club
Wednesday, 16 July –
South Africa vs New Zealand – Harare Sports Club
Friday, 18 July –
Zimbabwe vs New Zealand – Harare Sports Club
Sunday, 20 July –
Zimbabwe vs South Africa – Harare Sports Club
Tuesday, 22 July –
New Zealand vs South Africa – Harare Sports Club
Thursday, 24 July –
Zimbabwe vs New Zealand – Harare Sports Club
Saturday, 26 July –
Number 1 vs Number 2 – Harare Sports Club