Clark, the 2023 US Open champion, sank a four-foot eagle putt at the par-five 16th on the way to firing a level-par 70 at windy Shinnecock to stand on seven-under 203 after 54 holes.
The only golfer to lose a major with a lead of six or more entering the final round was Greg Norman at the 1996 Masters.
Scheffler, who would complete a career Grand Slam with a victory Sunday on his 30th birthday, fired a 69 — one of just two sub-par rounds on the day — to claim a share of second on one-under 209.
“It would be special,” Scheffler said.
“This tournament means so much to me. All I can do is go out there and try and execute.”
Scheffler opened with two bogeys for only the fourth time in 105 major rounds, but birdied four of his first seven holes on the back nine before a bogey at the par-three 17th and a closing par.
“We’ve been battling hard for a few days and I did a good job of keeping myself in the tournament,” Scheffler said.
“I’ll need a really nice round tomorrow if I’m going to try and catch Wyndham.”
Not since Lee Janzen in 1998 has a US Open winner been more than four strokes off the lead after 54 holes.
Joining Scheffler on 209 were fellow Americans Sam Stevens and Sahith Theegala, whose tap-in at 18 was his lone birdie of the day, plus South Korean Tom Kim, who shot 72 after bogeys on three of his first six holes.
“This was one of the biggest tests of patience I’ve ever had in a single round in my life,” Theegala said. “It was a grind.”
Whipping winds added to Shinnecock’s challenge, combining with dense rough and tricky pin positions to test the world’s top golfers.
Players are chasing a record US$4.5 million top prize from a purse matching the highest in major golf at US$22.5 million.
Clark opened with a three-putt bogey, tapped in for birdie at the par-five fifth but stumbled again to bogey eight after finding a greenside bunker with his approach.
He sank a five-foot birdie putt at 14 only to bogey 15 after a par putt miss from inside four feet, then answered with a four-foot eagle putt at the par-five 16th.
Clark, 32, is coming off a PGA Tour triumph four weeks ago in Texas and looking to wipe away the memory of his locker-smashing outburst after he missed the US Open cut at Oakmont last year.
After his bogey-bogey start, Scheffler parred through the front nine, birdied the 10th from just inside eight feet and reeled off three birdies in a row.
Scheffler’s birdie run, his longest in a US Open, started with a 65-foot birdie chip at 14 that prompted a fist pump.
“At that point over par for the tournament, you can feel like it’s slipping away,” Scheffler said.
“To steal one there was really nice.”
Scheffler followed with a 12-foot birdie putt at 15 and a tap-in birdie at the par-five 16th.
“It has been a battle all week,” Scheffler said.
“Got off to a slow start. It was nice to steal a few on the back nine. I’ve been fighting to stay in it all week.”
Second-ranked Rory McIlroy, who won his sixth major at April’s Masters, fired a 73 to stand 10 adrift on 213.
The six-time major winner birdied the fifth, sixth and seventh holes but had five bogeys in a back-nine collapse. (AFP)
