The defender rolled his left ankle during Sunday's 2-2 draw at Swansea and left the Liberty Stadium on crutches.
There were fears the 35-year-old may have suffered a fracture.
But an MRI scan on Tuesday morning showed that Terry suffered only minor ligament damage.
He will miss Chelsea's Premier League clash with Liverpool on Friday and next Tuesday's League Cup trip to Leicester, but will be hoping to face Arsenal at the Emirates on September 24.
In a post on his Instagram account, Terry wrote: "I have strained my ligaments in my ankle which means I will be out for 10 days.