Gibbs sprained his ankle during a training session in Pretoria before the team's departure for Sri Lanka but travelled with the squad and is now undergoing intensive treatment with physiotherapist Shane Jabaar.
South Africa hope he will be fit for the second Test in Colombo which starts on August 13 and the five-match, one-day series that follows.
"He is out of the first Test (starting on August 4) but the physio is working hard to get him going for the second Test," skipper Graeme Smith told reporters.
Left-handed opener Andrew Puttick, 23, was called up before the team's departure and will have an opportunity to stake his claim for his first cap during a three-day, warm-up match against a Sri Lanka Cricket President's XI starting on Friday.
