Spaun was on four-under after nine holes with compatriot Gary Woodland, the 2019 US Open winner, one back after reeling off three birdies in four holes.
Second-ranked McIlroy, who completed a career Grand Slam by winning the Masters in April, was in a pack on two-under with South Korean Kim Si-woo, Dane Niklas Norgaard and American Wyndham Clark, the 2023 US Open champion.
Spaun sank a 20-footer for birdie at 10 to start, dropped his third shot inside three feet at the par-five 12th to set up a birdie, then birdied the par-three 16th from just outside five feet to grab the solo lead at three-under.
At 17, he escaped the rough to sink an 11-foot birdie putt to reach four-under.
Woodland sank a 12-foot birdie putt at the second, a 25-footer to birdie the par-five fourth and a birdie putt at the fifth from just inside four feet to pull one adrift of Spaun.
McIlroy birdied the par-four 11th on a 27-foot putt and reached the green in two at the par-five 12th to set up an eight-foot birdie putt.
The five-time major winner from Northern Ireland could become only the seventh player to win the Masters and US Open in the same year.
Kim birdied the 11th from just inside four feet and the par-three 13th from just inside eight feet.
Clark had back-to-back birdies as well on putts of just inside seven feet at the third and 16 feet at the par-five fourth, where he blasted out of a fairway bunker.
Norgaard, making his US Open debut, birdied the 11th and par-three 13th on putts after about 16 feet each.
Sweden's Ludvig Aberg, last year's Masters runner-up in his major debut, birdied 10 from six feet and 11 from just inside three feet but stumbled to bogey the par-five 12th after finding rough and a greenside bunker.
Defending champion Bryson DeChambeau was one under through seven. He sent his tee shot on the first hole left into fan-trampled rough, landed his approach 18 feet from the hole, ran his birdie putt nine feet past the hole and missed a comeback par putt.
But DeChambeau birdied the fifth and seventh from just outside six feet to move under par.
American Maxwell Moldovan made the first US Open eagle at the par-four first, holing out from the fairway from 189 yards, then looking to the heavens as he smiled, but followed with back-to-back bogeys and a double bogey at the par-three sixth.