The 6-foot, 7-inch tall fast bowler had gone for a scan and it showed he has a disc strain in lower back, according to official team sources.
A final decision on the availability of Finn, who did not train with the team today, will be taken tomorrow morning, the sources added.
Finn had missed the opening two Test matches (Ahmedabad and Mumbai) of this tour after straining his thigh muscle on the opening day of the first warm-up game in Mumbai in late October.
He, then, played a practice game in Navi Mumbai for England's second string team where he took four for 50 to prove his match fitness and was eventually included in the third Test at Kolkata that the visitors won by seven wickets to take an unassailable 2-1 lead in the four-match series.
Finn took four wickets in that Test, three of those in the second innings when he dismissed Gautam Gambhir, Virat Kohli and Zaheer Khan to finish with impressive figures of three for 45, and play a key role in England's victory.
His absence in the fourth game here would be a major blow for the tourists in their quest to clinch a Test series in India after 28 years.
If Finn is found unfit, Tim Bresnan is likely to take his place in the playing eleven tomorrow.