The 27-year-old left-handed is back in India after spending time with his family in Melbourne.
Sunrisers coach, Tom Moody, tweeted, "we are hopeful it's within the next 10 days."
Dhawan suffered the injury during the Mohali Test against Australia last month. That was his debut game and he shot to prominence with a record-breaking 187, the fastest by a debutant in Test history.
"Hopefully I will be able to take the field soon. I just have to get the clearance from NCA. It will be soon, very soon," he had said before leaving for Australia.
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Dhawan's ODI debut, which took place three years ago, was a forgettable one.
Depsite the presence of stalwarts like Virender Sehwag and Gautam Gambhir at the top of India's batting order, Dhawan never lost hope of representing India in Test cricket.
It finally came after he was chosen ahead of the his senior Delhi teammates Sehwag and Gambhir, who had been dropped due to a prolonged run of poor scores.