Kyle Jamieson took three wickets as New Zealand pressed for a series-levelling win in the second test after leaving England with a record-breaking run-chase at The Oval on Saturday.
England were 182 for five at stumps on the fourth day, needing a further 281 runs to reach a mammoth target of 463 runs.
The most any side have made to win in the fourth innings in 149 years of test cricket is the West Indies' 418 for seven against Australia at St John's in 2003.
But stand-in England captain Joe Root was still there on 75 not out, having become just the second batsman after retired India great Sachin Tendulkar to score 14 000 test runs.
Not for the first time in his 165-test career, however, Root came in with England deep in trouble at 13 for two after towering fast bowler Jamieson took two wickets in five balls.
Root and Harry Brook (58) kept New Zealand at bay before New Zealand's first-innings bowling hero Matt Henry ended the Yorkshire duo's entertaining stand of 97.
Jamieson returned to have James Rew, one of three England debutants at The Oval, lbw on review shortly before stumps as he ended the day with fine figures of 3-37 in 14 overs.
The situation was tailor-made for the kind of counter-attacking innings associated with regular England captain Ben Stokes.
STOKES TO RETURN FOR NEXT TEST
But all-rounder Stokes, who made 95 for county side Durham on Saturday, is missing from this match after being dropped for breaking a team curfew following England's 115-run win in the first test at Lord's a fortnight ago.
Stokes will reportedly return for next week's series finale at Trent Bridge, but Root will have debutant Jordan Cox for company at the start of Sunday's play in south London as England try to stop New Zealand from levelling this three-match series at 1-1.
The fourth over of England's chase saw Emilio Gay (11) clip Jamieson to midwicket and four balls later the paceman had No 3 Jacob Bethell plumb lbw for a duck.
Root, the England captain before Stokes took over four years ago, needed just two runs to join Tendulkar in an exclusive club and a quick single off Henry took him to 14 000 test runs.
The 35-year-old, somewhat shyly given he was two not out, raised his bat as The Oval crowd applauded his achievement.
Tendulkar's 15 921 runs from 200 tests could be Root's next target.
But Root could only watch as Ben Duckett, on nine, recklessly tried to hook a Will O'Rourke delivery way above his head and spooned a simple catch to short midwicket.
New batsman Brook, showed better judgment in what to hit, with England needing to attack to have any chance of chasing down their imposing total.
He lofted Nathan Smith for a superb legside six and went down the pitch to drive Jamieson for four during a 33-ball fifty featuring 46 runs – 10 fours and a six – in boundaries.
Brook, however, could only manage seven more runs from his next 21 deliveries as the Black Caps regained control in the field and he eventually edged Henry, who had taken five wickets in England's first innings, to slip.
Root, who overturned being given out lbw on 44, went to his fifty by elegantly guiding Jamieson to the third man rope for an eighth four in 81 balls faced.
Earlier, New Zealand were dismissed for 362 in their second innings after resuming on 252 for three.
Henry Nicholls, in for retired New Zealand great Kane Williamson, added just two runs to his overnight 119, but Daryl Mitchell kept the runs coming with 68 after being dropped off Saturday's first ball.
ENGLAND: Ben Duckett, Emilio Gay, Jacob Bethell, Joe Root (capt), Harry Brook, James Rew (wkt), Jordan Cox, Jofra Archer, Josh Tongue, Matthew Fisher, Sonny Baker
NEW ZEALAND: Tom Latham (capt), Devon Conway, Henry Nicholls, Rachin Ravindra, Daryl Mitchell, Tom Blundell (wkt), Glenn Phillips, Nathan Smith, Kyle Jamieson, Matt Henry, Will O'Rourke