Jordan Spieth was tied for the lead at Carnoustie as he tries to become the first repeat victor in 10 years at the British Open.
Johnson (67 today) and first-round leader Kisner (70) are at six-under par through two rounds and holding the lead on a damper Carnoustie which still has not turned nasty with strong winds.
Woods, who shot 66 on Saturday, is paired with Molinari, two groups behind the final twosome of Spieth and Xander Schauffele. Spieth hit an 8-iron through a gap in the trees for a birdie-birdie start to the back nine, and he dropped only one shot - not four like he did on day one - over the four closing holes at Carnoustie for a 67.
"Judging by the conditions and what's going to be going on, I would imagine it's going to be more of a grind all day", said Kisner, who has been sharing a house with Spieth and several other American stars in Scotland this week.
Woods began his charge with a birdie at the fourth and picked up another shot at the sixth. "Michael said, 'No, you lay it up, and it's still a wedge to the front pin.' But walking to the tee, I was walking with Cameron [McCormick, his swing coach], and I thought, 'How about I just send it on No. 1?"
"It got brutal there the last four holes", Kisner told the Golf Channel.
It was the first time Woods had topped a major leaderboard since the Masters in 2013.
Americans have won the last five majors, but McIlroy has played well enough this week to believe he can claim a first major since 2014, when he did the double of Open and PGA Championship. "The intent was to get by the pot bunkers 60 yards short", Spieth said of the drive later, "but when we got close Bones [from US TV] said: "You're putting".
"I didn't want to be too far back if the guys got to 10-under-par today", Woods said.
Woods, who is playing in his 12th tournament this year after missing most of the past two seasons due to four back surgeries, opened the tournament with consecutive scores of par-71. They were last together three weeks ago when Woods presented him the trophy at the Quicken Loans National after Molinari shot 62 for an eight-shot victory.
When a 25-foot birdie putt trundled across the ninth green and into the cup things to get very interesting. He will try to become the first player since Padraig Harrington in 2008 to win golf's oldest championship in consecutive years. When he joined the leaders after another birdie at 14, he was tipping his hat to the crowds like it was 2004.
"I haven't played it in that much wind". This was his best performance in a major since he made the same score in the second round at Augusta seven years ago.