Play was delayed by a wet outfield caused by overnight rain and was called off just nine overs into the day's second session as rain returned.
Mominul Haque was batting on 40 with Mushfiqur Rahim on six at the other end when bad light forced players off the pitch and rain followed soon after.
More rain has been forecast for Saturday though a draw would be enough for India, who won the opening Test by 280 runs, to register a record-extending 18th consecutive Test series victory at home.
Earlier, home captain Rohit Sharma's decision to field after winning the toss sat well with Bangladesh counterpart Najmul Hossain Shanto, who wanted to bat first at the Green Park Stadium.
Once play started an hour behind schedule, India's new ball pair of Jasprit Bumrah and Mohammed Siraj maintained a probing line but could not make a dent in the Bangladesh top order.
Rohit introduced Akash Deep in the eighth over and the seamer struck with his third delivery.
Zakir Hasan had looked ill at ease right from the start and fell for a 24-ball duck edging the ball to Yashasvi Jaiswal, who took a low catch to enhance his reputation as a sharp slip fielder.
Shadman Islam (24) looked more comfortable in comparison, but Akash Deep returned to trap him lbw and cut short his stay.
The batter was originally adjudged not out, but Akash Deep convinced Rohit to review the decision, which was overturned after replays confirmed the ball would have gone on to hit the leg stump.
Mominul combined with skipper Shanto (31) in a 51-run partnership to try and rebuild the Bangladesh innings.
Shanto hit six fours in his knock but fell soon after the lunch break trapped lbw by India off-spinner Ravichandran Ashwin.
Shanto reviewed the decision but could not get it reversed.
Mominul's unbeaten knock included seven fours. India twice used the review system to try and get him lbw but the batter survived.
