Broad had scans on a bruised left heel after bowling only one over for Nottinghamshire in Leicestershire's second innings in the Specsavers County Championship Second Division innings victory this week.
But those scans have proved inconclusive and England believe a week's rest will solve the problem ahead of the opening Test, which starts at Lord's on July 6.
"Stuart has been assessed by the medics over the last 24 hours," Nottinghamshire head coach Peter Moores told the official club website. "They have ruled him out of the Kent game (which starts on Monday) for him to keep getting treatment, and then we'll wait to see how he responds to that treatment."
Broad hopes he will be fit for Nottinghamshire's Royal London One-Day Cup final against Surrey at Lord's on July 1 ahead of the Test opener with South Africa.
"I think he was really looking forward to playing in the Kent game," Moores added.
"He knows it's a big game for us in the season and he's been very much a part of the team for much of the last three months.
"Stuart has been around the block enough to know that's the way it is.
"He's got some pain and he's got a lot of cricket coming up so it needs sorting out.
"It's already started to settle and we'll have to see how it goes over the next four or five days."
Broad has had a busy time at Nottinghamshire this summer, playing five county championship matches and taking 16 wickets at 20.87.
The 30-year-old has also played 10 Royal London One-Day Cup matches, claiming 10 wickets.
Moores said: "He's desperate to play in the final, as you'd expect, and it's been fantastic to have him with us with the commitment he has shown to the club.
"We're all hoping it's going to be OK. He will do everything he can to be (ready), and now we just have to be patient and see how it turns out."
Broad is expected to be assessed in the nets at Lord's at the end of next week in a bid to prove his fitness for the cup final.
If he plays in that match without suffering a reaction it is likely he will be named in the squad for the first Test.
That would be a huge boost for England as Chris Woakes is already ruled out with a side injury and James Anderson has just returned after a groin strain.
