Ball, 26, was hurt during Notts' Championship match with Kent on Tuesday and scans revealed damage which is likely to keep him out for two weeks.
"We'll miss him for what will be a really tough game against Surrey," said Notts head coach Peter Moores.
England's four-match series with South Africa starts at Lord's on 6 July.
Ball has played three Test matches for his country, including two on the tour of India in December 2016, and played in all four of England's games in the Champions Trophy in June.
Nottinghamshire pace bowler Jake Ball will miss Saturday's One-Day Cup final and England's first Test against South Africa because of a knee injury.