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The 33-year-old injured his right thumb while playing for Sunrisers Hyderabad against Kolkata Knight Riders in the Indian Premier League playoffs last week.
"To determine the future course of injury management, Mr Saha will see a specialist and the BCCI Medical Team will continue to monitor his progress," BCCI said in a statement.
Saha's injury could pave way the for one of Parthiv Patel and Dinesh Karthik in the Indian squad.
Patel played two out of three tests in South Africa earlier this year, while Karthik has established himself as a regular member in the limited overs formats.
Afghanistan will make their test debut against India in the Bengaluru test starting June 14.