The 33-year-old wicketkeeper injured his right thumb while playing for Sunrisers Hyderabad against Kolkata Knight Riders in the Indian Premier League playoffs on May 25.
Saha will be replaced by Dinesh Karthik for the match starting June 14 in Bengaluru, when Afghanistan will make their debut in the five-day format after being granted full member status along with Ireland last June.
"He was under observation by the medical staff of the BCCI and the management has decided to give him adequate rest before the start of the England test series," the BCCI said in a statement.
"Saha's recovery period is expected to be around five to six weeks."
India will also be without regular captain Virat Kohli against Afghanistan. The batting mainstay opted to play for the English county championship side Surrey in June but was ruled out after injuring his neck while fielding during the IPL.
Saha has been India's regular gloveman since the retirement of former captain Mahendra Singh Dhoni in December 2014. He has played 32 tests and averages above 30 with the bat.
India begin their tour of England in July with three Twenty20 internationals and three one-day internationals before a five-test series starting in August.
(Reporting by Sudipto Ganguly; editing by Nick Mulvenney)
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