The Australian, who leads Martin Erlandsson (68), had eight birdies to improve to 11-under 131. Pelle Edberg (66), Steven Jeppesen (66) and Miguel Angel Jiminez (65) share third place at 9 under.
Garry Houston tied the course record at the Fontana Golf Club with a 63. He is at 7 under.
'I am happy with my steady play. I stayed very focused throughout,' Green said. 'I like the conditions here as the smooth greens are very similar to ours in Australia.'
Jiminez has not won in two years and failed to qualify for next week's U.S. Open.
'It is about time to win again,' Jiminez said. 'I hope to continue my performance over the weekend, and probably I can still talk about winning on Sunday.'
Erlandsson had seven birdies but added four bogeys.
Darren Clarke (70) and Colin Montgomerie (74) both missed the cut. Montgomerie double-bogeyed the par-4 third and 13th holes, and added two more bogeys on the 16th and 17th.
'I hit the fairways, but nothing happens. I can't score,' Montgomerie said.
Defending champion Markus Brier (73) also missed the cut.