Defending champion Paul Casey, right, talks with caddie John McLaren in Saturday's third round of the PGA Valspar Championship
The 41-year-old Englishman missed a 14-foot par putt at the par-4 18th and settled for a bogey and a three-under par 68.
He stood on nine-under 204 after 54 holes at the Innisbrook Resort Copperhead course in Palm Harbor, Florida.
Johnson, a winner at last month's WGC Mexico Championship, was next on 205 after a 67 with fellow American Jason Kokrak third on 206 after a hole-in-one on his way to matching the week's low round on 66.
Casey has struggled to defend titles and turn 54-hole leads into victories but he stands a chance of doing both on Sunday.
He has won only once in five tries, and never broken 70, when leading a US PGA event after 54 holes. The win came at the 2009 Houston Open with a bogey on the first playoff hole to beat American J.B. Holmes.
But the British star has taken back-to-back titles multiple times as an amateur. He won the 1999 and 2000 English Amateurs and three consecutive Pacific-12 Conference trophies while studying at Arizona State University.
Casey opened birdie-bogey after finding bunkers on each hole, added a six-foot birdie putt at the par-4 sixth and made back-to-back birdie putts from three feet at the par-5 11th and eight feet at the par-4 12th.
Casey stretched his lead to three strokes with a 10-foot birdie putt at the par-5 14th to reach 10-under.
At 18, Casey found a fairway bunker then dropped his approach into greenside sand and blasted out before rolling his par putt just off the right edge of the cup.
- 'Big birdie' -
Johnson fired his 14th consecutive round in the 60s, the longest such run of his career or by any tour player this season.
The World No. 1 sank a 33-foot birdie putt at the par-5 14th and curled in a tricky 10-footer for birdie at 18 to join Casey in the last pairing.
"It was a big birdie," Johnson said. "I'm in a good position going into the round tomorrow. I played a solid round. Missed a couple birdie opportunities but other than that I played really well."
Kokrak, seeking his first PGA crown after 197 starts over eight seasons, aced the par-3 15th with an 8-iron from 218 yards.
Kokrak, who began the back nine with three birdies on the first five holes, delivered only the third ace in tournament history at 15 when his ball bounced twice and rolled 25 feet into the cup, briefly giving him a share of the lead at seven-under.
"We had 194 to carry it into the downgrain part," Kokrak said. "It was into the wind. We knew if we flushed it, it would go right in and it was perfect."
Kokrak was trying to keep his emotions in check despite the joy of a hole-in-one.
"You can't get too high. You can't get too low," he said. "I made a good swing there. I got rewarded. You just have to make good shots."
Britain's Luke Donald, a former world number one who has battled a back injury, began and ended with birdies in shooting 70 to share fourth on 207 with American Scott Stallings.