India lost makeshift opener Irfan Pathan before reaching 18-1 at stumps on the fourth day in pursuit of what would be the highest-ever total to win a Test match in India.
Wasim Jaffer was unbeaten on four with nightwatchman Anil Kumble on eight.
The hosts, leading 1-0 in the series, now need 295 in 90 overs on the fifth and final day on Wednesday. The biggest target chased by a team to win a Test in India is 276, by the West Indies at Delhi in 1987.
Pathan opened the innings on Tuesday in place of unfit Virender Sehwag, who was off the field for most of the day with back spasms. Sehwag can now bat only after the fall of the fifth wicket.
England, leading by 121 runs on first innings, were earlier dismissed for 191 in their second innings despite stand-in captain Andrew Flintoff's 50.
Leg-spinner Kumble finished with 4-49 and off-spinner Harbhajan Singh with 2-40.
Flintoff survived a stumping chance on 14 and two other close calls while playing a crucial knock under pressure. He stepped out to drive Harbhajan and misread the line, but wicket-keeper Mahendra Dhoni failed to gather the ball.
When Dhoni finally stumped Flintoff off Kumble, the England skipper had completed his fourth successive half-century.
Flintoff, who made 70 and 51 in the second Test at Mohali and 50 in the first innings of this match, hoisted Harbhajan into the stands for a six and also hit six fours.
Debutant Owais Shah (38), Paul Collingwood (32) and nightwatchman Shaun Udal (14) all showed resolution at a time when England badly needed runs.
India held a pair of splendid return catches to try to stay in the match, with Kumble dismissing Ashes hero Kevin Pietersen and Harbhajan accounting for the well-set Collingwood.
Kumble, bowling round the stumps, moved swiftly to his left to hold an uppish drive to remove Pietersen for seven. Harbhajan took a left-handed diving catch on his follow-through.
Collingwood had played a key role in steadying his team with a 66-run stand for the sixth wicket with his captain, who offered hope to the bowlers and fielders during his 146-ball knock.
Flintoff had a close shave before he had opened his account but silly-point fielder Yuvraj Singh failed to hold a forcing shot off Kumble. England hearts again skipped a beat when a mistimed Flintoff drive fell short of Kumble.
The Indian spin duo posed problems of turn and bounce but England still managed to set a tough target.
They batted cautiously, adding 54 in the first session and 53 in the second. But they made India work hard for success, losing two wickets in the morning and just one in the afternoon.
Overnight batsman Udal was the first to go but not before frustrating India for more than an hour. He was caught low at second slip by Jaffer off left-arm seamer Pathan.
Shah, who scored an impressive 88 in the first innings, batted more than a session before being caught short of the crease by Sachin Tendulkar's throw to the wicket-keeper.