Ollie Robinson's two wickets in the brief passages of play that were possible at Lord's mean England need five more wickets to win when play is set to resume on day four on Sunday, live on Sky Sports.
New Zealand will have work to do on 55-5 in their second innings, needing another 199 runs to secure just their second Test win at Lord's.
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Only 9.4 overs were possible on the third day, which followed a whirlwind 17 wickets falling on the first two days' play.
New Zealand had resumed on 36-3 when play finally got under way at 1pm. It came after the entire morning session was lost to a combination of sharp showers but also the outfield needing time to dry out despite prolonged periods of bright sunshine.
It led to a bizarre quirk of the playing conditions where an early lunch was taken at 12.20pm, which coincided with the brightest and driest conditions of the day and caused much frustration among spectators and pundits.
Sky Sports presenter Ian Ward called the situation "ludicrous" as he referenced how golf tee-times are altered and the timing of F1 races shifted in the event of wet weather.
"There is heavy rain forecast from about 1pm, certainly from 3pm, and no action is going to take place while the sun is shining. That is cricket in a nutshell sometimes, I'm afraid," he said.
When play did finally start, a couple of brief interruptions followed before Robinson snared the wickets of Rachin Ravindra and Daryl Mitchell in successive overs.
Ravindra (8) had his off-stump clipped by one that nipped away before Mitchell (0) was trapped leg before and fell the wrong side of an umpire's call on review to leave the tourists on 53-5, still 201 runs shy of victory.
But rain soon returned as Tom Blundell (2no) joined Devon Conway (19no) at the crease and would remain until late into the afternoon when the decision was finally taken at 5.29pm to abandon the day's play.
Watch day four of the first Test between England and New Zealand live on Sky Sports Cricket and Main Event from 10.15am on Sunday. Not got Sky? Get Sky Sports or stream with NOW.
