India captain Shubman Gill became the first batsman in test history to make scores of 250 and 150 in the same match before Mohammed Siraj and Akash Deep again struck with the new ball as the tourists pressed for a series-levelling win at Edgbaston on Saturday.
Gill followed his first-innings 267 in the second test with another superb knock of 161 off just 162 balls.
He eventually declared India's second innings on 427-6 after tea on the fourth day.
That left England needing a mammoth 608 runs to go 2-0 up in this five-match series.
No side in 148 years of test cricket have made more to win in the fourth innings than the West Indies' 418 against Australia at St John's in 2003.
England's corresponding record is the 378 they made against India at Edgbaston three years ago.
But at Saturday's close in Birmingham they were 72-3, needing a further 536 runs on the final day to achieve a win that would be remarkable even by this team's 'Bazball' standards.
"Nothing is impossible," England batting coach Marcus Trescothick told the BBC.
"But 536 is a lot of runs... If we were to chase that down then it would be an epic day's play."
Harry Brook, who helped keep England afloat in their first innings with 158 in a total of 407, was still there on 15 not out.
Siraj, who led India's attack with 6-70 in the first innings, struck an early blow Saturday when he removed Zak Crawley for a duck as a loose drive went straight to backward point.
Deep, given the unenviable task of replacing the rested Jasprit Bumrah, the world's No 1 ranked fast bowler, then carried on from his four-wicket haul first-time around by bowling Ben Duckett (25) with a nip-back ball.
He then clean bowled Joe Root for just six as the world's No 1-ranked test batsman was undone by a superb full-length delivery that moved away late.
"That's a pleasing sign from Akash, he is an attacking bowler that asks questions, bowling at the stumps a lot," India bowling coach Morne Morkel told reporters.
The former South Africa paceman added: "Top-quality player, Joe Root, and to dismiss him in that fashion just shows the quality of Akash."
Earlier, by taking his tally in this match to 430 runs, the 25-year-old Gill became only the fifth batsman to score 400 or more runs in a single test.
Gill, thrust into the captaincy following Rohit Sharma's shock retirement from test duty in May, has scored three hundreds in his first four innings as skipper following his 147 during India's five-wicket defeat in the first test at Headingley.
He was ably assisted earlier Saturday by Rishabh Pant, who made England pay dearly for dropping him twice, including a poor miss by Crawley at mid-off when the wicketkeeper had made just 10.
Pant cashed in with a typically dashing 65 during a fourth-wicket partnership of 110 in just 103 balls with his skipper.
TONGUE LASHING
But Gill, often a classical stylist, showed he could bat just as aggressively as Pant by hitting fast bowler Josh Tongue for three consecutive boundaries.
A superb hooked six was followed by a swiped four over mid-on and a scorching pull that England captain Ben Stokes could not stop.
Gill, batting in the No 4 slot previously occupied by India star Virat Kohli, who retired just days after Rohit, hooked another six off Tongue.
Pant's innings ended when, trying to smash offspinner Shoaib Bashir, he lost control of his bat, which flew to midwicket while the ball went to deep mid-off – where Duckett made no mistake with the catch.
Pant faced just 58 balls, including eight fours and three sixes.
After tea, Gill piled on the runs with a brilliant straight six and two fours off consecutive balls from Chris Woakes, before slog-sweeping part-time offspinner Root for six to go to 150 in superb style.
Gill's lates masterclass finally ended when he spooned a simple return catch to Bashir, having hit 13 fours and eight sixes.
There were crowd chants of "Boring, boring India" as the tourists batted on before Gill finally called a halt, with Ravindra Jadeja 69 not out.
ENGLAND: Zak Crawley, Ben Duckett, Ollie Pope, Joe Root, Harry Brook, Ben Stokes (capt), Jamie Smith (wkt), Chris Woakes, Brydon Carse, Josh Tongue, Shoaib Bashir
INDIA: Yashasvi Jaiswal, KL Rahul, Karun Nair, Shubman Gill (capt), Rishabh Pant (wkt), Nitish Kumar Reddy, Ravindra Jadeja, Washington Sundar, Akash Deep, Mohammed Siraj, Prasidh Krishna