England were dismissed in their first innings for 476, with Joe Root breaking out of a prolonged slump with a majestic 226.
New Zealand captain Kane Williamson and senior batsman Ross Taylor loom as the key wickets England need to capture to win the second test as the hosts finished the fourth day on 96-2 at Seddon Park in Hamilton on Monday.
Williamson was on 37, while Taylor was 31 as New Zealand head into the final day still facing a five-run deficit after England were dismissed in their first innings for 476, with Joe Root breaking out of a prolonged slump with a majestic 226.
The England captain had resurrected his side's innings with a 177-run partnership with Rory Burns (101) on Sunday and 193 with Ollie Pope, who had been in sight of his first test century before he fell for 75 as they chased quick runs before tea.
New Zealand won the first match of the two-test series at Bay Oval in Mt Maunganui by an innings and 65-runs.
Heavy rain, however, is forecast for Hamilton after lunch on the fifth day and England will need to run through New Zealand's batting early on to seal victory.
Root's third test double century had put his side into a strong position, which was only made stronger when Sam Curran trapped Jeet Raval in front for a two-ball duck.
Raval did not review the decision, despite being asked by fellow opener Tom Latham if it was worth it, but technology suggested he had got an inside edge on the ball before it cannoned into his pad.
Latham followed shortly after for 18 when he nicked a Chris Woakes delivery to Root at first slip to leave New Zealand struggling at 28-2, before Williamson and Taylor drove their side through to stumps.
England's innings had been dominated by Root, who brought up his 17th test century on Sunday to break out of his prolonged slump.
He had not achieved three figures in a test since scoring 122 against West Indies in St Lucia in February - a stretch of 15 innings - and was averaging 27.10 this year prior to the Seddon Park match.
Root, who had resumed on Monday on 114, brought up his 200 after lunch with a quick single into the offside that left Pope having to dive to make his ground, although the throw at the stumps was wayward.
He eventually was caught in the deep for 226 as he tried to increase the scoring rate before tea, falling four balls after Pope had been caught attempting to do the same thing.
Root moved inside the top-10 test run scorers for England with his innings, advancing to 7282 runs at an average of 48.54, the highest in the list.
Neil Wagner finished with 5-124 as he ran through the tail.