Rahul, who has played 50 tests for India and scored eight centuries, missed the second test of the five-match series due to an injury.
"Mr. Rahul has reached 90 per cent of match fitness and is progressing well under the supervision of the BCCI medical team," the BCCI said in a statement.
"He will continue his recovery process at the National Cricket Academy (NCA) in Bengaluru to be completely match-fit for the fourth and fifth test."
Rahul is the second major Indian batsman to be out of action during the third test, after Virat Kohli pulled out of the series due to personal reasons.
England spinner Jack Leach will also miss the rest of the series due to a knee injury.
Padikkal, who has replaced Rahul in the squad, has represented India in T20 cricket but is yet to make his test debut.
India and England are tied 1-1 in the series as the third test gets underway in Rajkot on Thursday.
