The 26-year-old Shami played the first three tests of the series but missed the fourth in Mumbai with a sore knee and has been advised rest and rehabilitation.
"He will be going to the National Cricket Academy, Bengaluru to start his recovery programme," the Indian cricket board (BCCI) said in a statement on Sunday.
Fast bowler Ishant Sharma, who has not played in the series so far and was released for his marriage ahead of the fourth match, is expected to return to the squad for the Chennai test.
The Indian selectors had named paceman Shardul Thakur as a back-up for Shami for the Mumbai test.
Saha suffered a hamstring tendon injury in the second test at Visakhapatnam and did not play in Mohali and Mumbai. He is undergoing rehabilitation programme at the NCA, the BCCI added.
Wicketkeeper Parthiv Patel replaced Saha for the hosts, who are up 2-0 in the five-match series.
(Editing by Amlan Chakraborty)
