He posted a 16-under 200 total with one round left as he seeks his maiden PGA Tour win in his 88th start.
Conners shot 67 to move into second place at 14-under, with American Tyler McCumber next at 12-under.
"It was playing really difficult. The wind made things challenging but I hit the ball solidly all day," said Conners, who will play in the last group out on the final day for the second time this month.
He led after three rounds at the Valspar Championship two weeks ago before fading with a closing 77 to plunge to equal 16th.
Conners hopes the experience will help.
"I know it's important to stay relaxed, slow things down a little bit, make good decisions and just trust my game," he said.
"I've got a lot of confidence right now and been playing well for quite a while."
(This version corrects margin of lead to two strokes)
(Reporting by Andrew Both in Cary, North Carolina, editing by Ed Osmond)
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