He started the match but lasted just 18 minutes before being replaced - despite having already slotted two conversions and a penalty.
"We were monitoring him during the game and he was struggling to kick and run, so we made the call to take him off," Springbok team doctor Craig Roberts said at a press conference in Edinburgh on Monday.
"Clinically it looks like a facet-joint irritation, which usually takes two to three days to settle. It's already looking much better today and our physiotherapists will work hard to sort him out.
"Luckily we've got a long week to get him ready and we're hoping to have him available for selection this weekend. Based on how he's responded so far, it's looking good."
The team to face Scotland will be announced on Thursday. Should Steyn not be available it is likely Pat Lambie will move from fullback to flyhalf as his replacement.
(Reporting by Nick Said in Cape Town; Editing by Tony Goodson)
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