Finn was injured during England's first warm-up match of the trip and bowled just four overs. He is now expected to undergo a further scan in Mumbai on Wednesday to extent of the problem.
Finn had been tipped to be recalled for the second Test to bring a cutting edge to England's pace attack which struggled to make an impression in the opening Test in Ahmedabad.
James Anderson, Stuart Broad and Tim Bresnan managed just one wicket between them and were considerably out-bowled by India's Zaheer Khan and Umesh Yadav.
There is still a strong chance that England will make changes to their bowling attack with Monty Panesar tipped for a call-up, while there will be an enforced changed in the batting after Ian Bell flew home for the birth of his child although he did not make it home in time for the delivery.