JERSEY CITY, August 24, 2013 (AFP) - Kuchar, who led after the second round concluded on Saturday morning, fired a one-under 70 in round three.
He and Woodland finished 54 holes at 12-under par 201.
Another marathon day on Saturday got the tournament back on schedule for the first time since weather delays totalling nearly six hours on Thursday.
Kevin Chappell moved into contention by firing a course-record, nine-under 62 in the third round. That vaulted him into third place at minus-11.
Playing with a sore back, Tiger Woods shot a two-under 69 and was four shots back of Kuchar and Woodland.
England's David Lynn also posted a 69 Sunday afternoon and joined Woods in a tie for fourth at eight-under 205.
Defending champion Nick Watney shared 13th place after a third-round 69 for 207.
Despite the back pain, Woods climbed up the leaderboard with four birdies in his final 11 holes.
"It starts off great every day, and then it progressively deteriorates as the day goes on," said Woods, who had trouble bending over at times on the course. "Hopefully tomorrow it will be one of those days again and fight through it and see if I can win a tournament."
Woods was pleased with the way he toughed it out Saturday. He hit all but two fairways and 13 of 18 greens.
He also adjusted his swing slightly to put less pressure on his sore back.
"I mean, once you start off with it and then you keep playing on it, practicing, warming up, I'm loading it pretty good," said Woods, adding he feels the sore back is a result of sleeping on a hotel mattress that is too soft for him.
"It's not like I go out there and puff it around. I kind of go at it a little bit."
Leading scores on Saturday after the third-round of The Barclays, $8 million opening event in the US PGA Tour playoffs (USA unless noted, par-71)
201 - Matt Kuchar 66-65-70, Gary Woodland 69-64-68
202 - Kevin Chappell 68-72-62
205 - Tiger Woods 67-69-69, David Lynn (ENG) 71-65-69
206 - Kevin Streelman 70-68-68, Jordan Spieth 70-68-68, Bubba Watson 68-70-68, Jim Furyk 70-66-70, Justin Rose (ENG) 68-68-70, Jonas Blixt (SWE) 69-67-70, Rickie Fowler 71-64-71
207 - Nick Watney 68-70-69, Aaron Baddeley (AUS) 69-72-66, Rory McIlroy (NIR) 71-65-71, Sergio Garcia (ESP) 70-66-71, Adam Scott (AUS) 69-66-72, Webb Simpson 67-66-74
208 - Daniel Summerhays 70-69-69, Matt Every 67-72-69, Ryan Moore 67-72-69, John Huh 73-64-71, Brendon de Jonge (ZIM) 67-69-72, D.A. Points 70-72-66, Greg Chalmers (AUS) 73-69-66