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Auckland: Williamson picked up two wickets in the last session on Day 4 to leave England struggling at 90/4 at stumps, chasing 481 against New Zealand at Eden Park on Monday.
Alastair Cook was out for 43 and Jonathan Trott 37. The pick of the New Zealand bowlers was Kane Williamson who has two for five.
England lost Nick Compton (2), Jonathan Trott (37), Alastair Cook (43), and Steve Finn (0) before stumps, suffering its most bitter blow when Cook fell in sight of the close of play.
New Zealand declared their second innings at 241 for six with Peter Fulton making 110 his second hundred in match and skipper Brendon McCullum not out 67.
England were in immediate trouble in their run chase with Nick Compton out for two and they went to tea at 45 for one.
There are three sessions remaining in the Test with the series locked at 0-0 after the first two, rain-affected, encounters were drawn.
Only four teams have scored more than 400 in the fourth innings to win a Test, with the record held by the West Indies who reached 418 for seven to beat Australia in Antigua in 2003.
Fulton set up the mammoth New Zealand total with his second century of the match, and of his Test career, scoring 110 in the second innings to follow his earlier 136.
It was only the fourth time a New Zealand batsman has scored back-to-back centuries in a Test and the first time the feat has been achieved since Andrew Jones produced a double against Sri Lanka 22 years ago.
The 34-year-old opening batsman showed no sign of nerves in the nineties and reached his milestone driving Stuart Broad over his head for six, the fifth time he had cleared the ropes in an innings that also included 14 fours.
New Zealand resumed the day at 35 for three with the aim of posting a forbidding total, cracking 141 off 26 overs in the morning session.
Dean Brownlie, who had done a sterling job with Fulton to resurrect the innings after New Zealand were eight for three early on, was the only wicket to fall before lunch.
He fell to an exceptional catch by Ian Bell who ran 25 metres from mid-on to clasp the ball in a desperate lurch which left him sprawled on the ground.
Brendon McCullum joined Fulton in the middle and they belted 117 off 101 balls before Fulton was caught by Joe Root off Steven Finn.
When BJ Watling was dismissed for 18, McCullum was unbeaten on 67 and the captain declared the innings closed.
Brief scores
New Zealand 443 (Fulton 136, Williamson 91, Finn 6-125) and 241 for 6 dec (Fulton 110, McCullum 67*)
England 204 (Prior 73, Boult 6-68, Southee 3-44) and 90 for 4 (Cook 43, Williamson 2-5)
Status England need 391 runs to win
