Furyk, playing early, had a bogey on the ninth hole -- his final one of the day -- and Scott, playing late, had a bogey at 16 leaving both at 5-under-par 65.
Ben Hogan set the record of 64 at Oak Hill Country Club in Pittsford, N.Y., near Rochester in 1942 and Curtis Strange tied that mark in 1989.
Furyk and Scott are one stroke ahead of David Hearn, who also had a bogey on his final hole Thursday, and Lee Westwood, who was bogey free en route to a 4-under 66.
A chorus of six players came in at 3-under with U.S. Open winner Justin Rose just missing the group when he had an 18th-hole bogey and ended up at 2-under.
Scott, the Masters champion, used a birdie string over Nos. 4-8 to pull even with Furyk at 5-under. Scott got to 6-under with a birdie at 14 before retreating with the 16th-hole bogey.
Furyk was 6-under through 16 holes but had a final-hole bogey at the ninth hole but took a 5 on the par-4 hole.
Defending champion Rory McIlroy was 3-under early in his round but dropped back to 1-under in a round that saw five birdies and four bogeys.
The world's highest-ranked players each closed with double bogey.
No. 1 Tiger Woods was 2-under through 12 holes but had a bogey at the fourth and a 6 at the par-4 ninth to close his round.
Phil Mickelson, No. 2 in the rankings after winning the British Open July 21, had two double bogeys. The first was at the fourth hole after he drove the ball out of bounds and the second was at 18 after he failed to get the ball out of trouble on the right.
Those results left Woods and Mickelson 1-over for the round.
Mickelson was grouped with Scott and Rose as the winners of the year's first three major tournaments.