Kuchar carded an eight-under 64 on the Jack Nicklaus Private Course at PGA West to claim the halfway lead in the tournament, while Palmer went three shots better at the same layout with a sensational round of 61.
Palmer, who started on the 10th tee, looked on track for one of the greatest rounds in history after taking just 27 shots on the back nine to head back out into the country on nine under.
He began his hot streak with an eagle-two on the par-four 12th and then added three successive birdies before posting another eagle on the par-five 16th and then back-to-back birdies on the 17th and 18th.
The 38-year-old American???s front nine started with another birdie on the first hole to equal a PGA Tour record of eight straight holes with either an eagle or birdie and also become the first man to play eight consecutive holes in 10 under par.
Bogeys at both the second and third checked his charge and dampened his hopes of breaking 60, but he got back on track with birdies at the fourth and sixth and another birdie followed at the eighth.
However, he could only manage a par four on the ninth after missing a five-foot birdie putt and had to settle for a share of the course record of 61.
Kuchar also enjoyed playing the back nine as he recorded five birdies, but he then produced a mixed bag on the front nine with two birdies, an eagle at the par-five fourth and a bogey at the sixth to move to 15-under for the tournament.
The 36-year-old is one shot clear of first round leader Michael Putnam, who shot a five-under 67 at Arnold Palmer Private Course at PGA West, and another American Bill Haas, who carded a nine-under 63 at La Quinta Country Club.??
But it was the Jack Nicklaus Private Course where most of the low scores were posted as American trio Nick Watney (64), Justin Thomas (63) and Scott Pinckney (67) all moved into a share of fourth place on 13 under.
Scotland???s Martin Laird impressed on the Arnold Palmer Private Course with a six-under 66 to sit joint-14th on 10 under.
Defending champion Patrick Reed and Italian Francesco Molinari both slipped back to nine under after rounds of 70 and 71 respectively, but Phil Mickelson, who is back in action this week after a lengthy lay-off, will have been encouraged by his six-under 66 on the Nicklaus course which moved him up to seven under.