Reigning U.S. Open champion Johnson, who can claim the top ranking with a win should incumbent Jason Day tie for third or worse, made six birdies and one bogey to tie for second behind Saunders on the classic Riviera layout in Los Angeles.
"Obviously, it would be great to get there but I'm focussed on this week and this golf tournament, each shot that I'm hitting," said Johnson, who is in a five-way tie for second.
Saunders, grandson of the late Arnold Palmer, made seven birdies in a bogey-free 64 as he seeks his maiden PGA Tour win in his 79th start.
"I just want to try and give myself the position to win any tournament out on tour," said Saunders, whose famed grandfather won the Los Angeles Open three times.
"I'm still looking for my first win. If it could come at a place like this it would be incredibly special," the 29-year-old added.
Joining Johnson at five-under were J.T. Poston, a qualifier who holed out from the fairway to eagle the par-five first hole, Daniel Summerhays, Brett Stegmaier and Cameron Percy.
Among a pack of eight posting 67s are Ireland's Padraig Harrington and crowd favourite Phil Mickelson, who displayed his short game magic to the delight of fans.
Mickelson began the round at the 10th and surged up the leaderboard with an eagle at the par-five 17th after banging his 317-yard approach off the greenside grandstand into deep rough.
But the big lefty turned trouble into triumph when he hoisted up a high, soft pitch that found the bottom of the cup.
World No. 5 Hideki Matsuyama of Japan, who also has a chance at grabbing the number one spot with a win, was one-under through 16 holes when play ended.
Australia's Day was tied for 74th on even par after 16 holes, two strokes better than defending champion Bubba Watson, who had played 17.
(Reporting by Larry Fine in New York; Editing by Frank Pingue/Peter Rutherford)
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