England were 260 for eight in their second innings as they stretched their lead to 233 runs
Jos Buttler's well-judged 69 helped leave India facing a record run-chase if they are to win the fourth Test against England at Southampton and so level the series.
England were 260 for eight in their second innings, a lead of 233 runs, at stumps on Saturday's third day.
The most India have made to win in the fourth innings of a Test in England is 174 at The Oval in 1971. That match also marked their first Test-match win on English soil.
Sam Curran, fresh from his Test-best 78 in the first innings, was 37 not out after adding 55 for the seventh wicket with Buttler.
Adil Rashid (11) was caught behind off a Mohammed Shami leg-cutter off what became the last ball of the day.
The paceman led India's attack with three wickets for 53 runs from 13.5 overs.
Last week saw Buttler display a well-judged mixture of aggression and defence during his maiden Test century, albeit one made with India well on the way to a 203-run win in Nottingham that left them 2-1 down in this five-match series.
Those same qualities were on show in Southampton, where Buttler made 85 on his Test debut, also against India, four years ago.
They were needed after England slumped to 92 for four — the 33rd time in 64 Test innings they had been four wickets down for 100 or fewer runs.
Shami took two wickets in two balls separated by the lunch break before running out England captain Joe Root for 48 with a direct hit.
England, however, have not lost successive home Tests since suffering back-to-back defeats by South Africa in Leeds and Birmingham in 2008.
"We've plenty to work with," Buttler told Sky Sports. "Anything above (a lead of) 250 would be a good number. It can be challenging out there. I think all of our bowlers will be a threat tomorrow (Sunday).We feel like we've got lots of bases covered."
But Cheteshwar Pujara, who made a superb maiden Test century in England on Friday, said an India side led by Virat Kohli had the batting firepower needed to chase down a stiff target.
"Looking at the pitch it has slowed down a bit," Pujara said after Saturday's close. "It looks like it is slightly easier to bat. And we have got lot of experience playing in such conditions back home."
England resumed Saturday on six without loss.
They were then 21 runs behind, after Pujara's 132 not out had guided India to a first-innings total of 273.
Alastair Cook, England's all-time leading Test run-scorer, and fellow left-handed opener Keaton Jennings both needed runs.
But Cook fell for 12 when, drawn into a loose drive off fast bowler Jasprit Bumrah, he got an edge that second slip KL Rahul clung onto at the third attempt.
England were soon 33 for two.
Moeen Ali, promoted to number three instead of Root, became the latest left-handed batsman to fall victim to Ishant Sharma from around the wicket when a low edged drive saw him caught superbly by Rahul for nine.
It had looked as if Jennings, under pressure for his England place after a first-innings nought, would bat through the morning until Shami had him lbw for 36 with what became the last ball before lunch.
England had decided to play Jonny Bairstow as a specialist batsman even though he was nursing a broken finger suffered in the field at Trent Bridge last week that meant he could not keep wicket.
It was a move that did not pay off, with Bairstow suffering a second nought in three innings when he optimistically tried to drive a Shami inswinger off the first ball after lunch.
Root was settling in when, called for a needless single by Ben Stokes, he failed to beat Shami's direct hit from mid-on.
Stokes, as he'd done at Nottingham, showed plenty of grit in a 30 off 110 balls that ended when he edged off-spinner Ravichandran Ashwin to Ajinkya Rahane at slip.
Buttler achieved a better tempo, going to fifty by cover-driving an Ashwin full-toss for his fifth four in 96 balls.
India took the new ball at a sun-drenched Ageas Bowl with England 223 for six off 82 overs.
It had the effect they wanted when Buttler, aiming across the line, was lbw to Sharma.
But 20-year-old left-hander Curran struck Bumrah for two fours in as many balls, including an elegant cover-drive.