With three days still left to play, India could dictate the course of the game after skittling England out for 161, with Hardik Pandya taking a stunning 5-28 in six overs - the pace-bowling all-rounder's maiden five-wicket Test haul - on Sunday.
England openers survived nervy nine overs in the last session on Day 3.
The record fourth-innings score to win a Test is 418, by the West Indies against Australia in 2003.
The highest in their history dates back nearly 90 years, to the 332 for seven they made in the 1928/29 Ashes in Melbourne.
The result will not only keep the contest alive - England will lead 2-1 with two matches to play - but now brings into focus the hosts batting, a role reversal of the Lord's Test after which India's batters were widely criticised.
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Indian batsman Cheteshwar Pujara admitted that he was under pressure to score runs and credited his county stints for providing him the confidence to bat in English conditions and overturn his poor form during the third Test here. He stroked 10 boundaries in his 197-ball essay. The Indian skipper scored his 23rd test century in style and as soon as he raised his bat proudly to the crowd, Anushka Sharma got up in joy and clapped for his achievement.
There was to be no denying him his century and an edged boundary off Chris Woakes, his 10th four from 191 balls faced in almost five hours at the crease, saw him to a 23rd Test hundred.
England's Jonny Bairstow is the second highest run-scorer in the series with 206 runs.
He was then sent for an X-ray on the injury to his left middle finger. However, an European Central Bank spokesman announced the likelihood is that Bairstow will be able to bat.
England saw out the morning session without sustaining further damage as an outstretched Rishabh Pant dropped Buttler on one after a misstep, an error that would prove costly after lunch. This is the first time in Test history that two players from the same team have picked up six catches apiece in a match.
If at all, the viewers are not able to catch the live action on their TVs, they can watch India vs England 3rd Test Match live through online streaming. Kohli has shown the class to deal with it. "Kohli is the best player in the world, his technique he is constantly improving and learning about and that's something we can all learn from".