Scheffler, who would complete a career Grand Slam with a victory Sunday on his 30th birthday, lipped out on a six-foot par putt at the par-three second hole after finding a bunker.
World number one Scheffler, a four-time major champion, and second-ranked McIlroy, a six-time major winner, began the day seven strokes behind leader Wyndham Clark, the 2023 US Open winner.
Whipping winds changed directions to add to Shinnecock's challenge, combining with dense rough and tricky pin positions to test the world's top golfers.
American Clark, the 36-hole leader at seven-under par 133, was set to start at 3:45 p.m. (20:45 CET) with England's Matthew Fitzpatrick, the 2022 US Open champion who shared second on 137.
South Korean Tom Kim and Americans Xander Schauffele and Sam Stevens were also four adrift, with American Collin Morikawa another stroke behind.
Not since Lee Janzen in 1998 has a US Open winner been more than four strokes off the lead after 54 holes.
Players are chasing a record $4.5 million top prize from a purse matching the highest in major golf at $22.5 million.
Scheffler plunked his first two shots into the left rough at the opening hole and sent his third into a greenside bunker on the way to a bogey.
McIlroy found the left rough off the first tee, then a greenside bunker, but got up and down for par.
Argentina's Emiliano Grillo proved low scores were available, reeling off four consecutive birdie putts to close the front nine, the last and longest from just inside 25 feet, to reach one-under overall.
England's Tommy Fleetwood eagled the par-five fifth to reach level par.
There were only 10 players under par through the first two rounds.
Clark, 32, won his fourth PGA Tour title four weeks ago at the CJ Cup Byron Nelson event. He has called this week's event a chance for redemption after smashing a locker at Oakmont following a missed cut in last year's US Open.
Fitzpatrick, 31, has won three times on the PGA Tour this year, the Valspar Championship in March, the Heritage tournament in April and the tour pairs event in New Orleans with brother Alex.
Schauffele, 32, won the 2024 British Open and PGA Championship and has seven top-10 finishes in nine prior US Open starts, never finishing worse than 14th. His best result was a share of third in 2019.
Kim, who turns 24 on Sunday, has not won in nearly three years. He suffered a right ankle ligament injury in July 2023 but fought through it to finish second in the British Open.
Stevens, 29, is making his eighth major appearance and has never finished in the top 20. He has never won a PGA Tour title.