Australia's Day carded a three-under 68 at the Ridgewood Country Club and American Furyk (69) joined him atop the leaderboard at nine under.
It came as Scott, the overnight leader alongside Cameron Tringale, struggled in his title defence.
World number two Scott (75) holed three bogeys and a double bogey to go with his birdie at the 13th.
He slipped to a tie for 23rd at four under, while Tringale (72) is two shots behind the leaders.
Day needed a strong finish to surge into contention after a double bogey on the par-five 13th.
The 26-year-old holed three birdies – and a bogey – on the front nine and went three under for the day at the 10th.
But Day took seven shots at the 13th, damaging what had been an impressive start to the round.
He responded well with three birdies on his final four holes.
Furyk was far more consistent as he produced birdies at the 13th and 16th holes to go with 16 pars.
American Hunter Mahan (68) was left to rue a bogey at the 18th, sitting at eight under and outright third.
A group of seven Americans are at seven under.
Morgan Hoffmann (66), Matt Kuchar (68), Kevin Na (70), Bo Van Pelt (70), Kevin Chappell (71), Brendon Todd (71) and Tringale (72) are among them.
Englishman Justin Rose (70) is still in contention at five under, a shot ahead of in-form world number one Rory McIlroy (70).