Green fired a 3-under 69 on Thursday to join a quartet of players to be just one off leader Stephen Ames.
The left-hander played the first nine in 1-under 35, but came home strong with birdies on the 10th, 14th and 17th. A lone bogey on the par-four 15th was the only blotch on his card.
Ames recorded five birdies against one bogey as the 60-year-old Canadian seeks his third win of the calendar year - he won the Chubb Classic in February and the Mitsubishi Electric Classic in April.
He is trying to join South African Ernie Els for the most victories by a Champions Tour player this season.
Ames is bidding for his ninth win on the Champions Tour and first major.
"Well, hit the ball nicely," Ames said of his round. "Hit some good shots and I missed a lot of putts out there, unfortunately, for birdies. But I think overall, the pace of play kind of hindered us a little bit coming down at the end there. So that kind of threw me off-balance.
"Overall this is a good test here this week for us. So I quite enjoyed the 68."
Ames holds a one-shot lead over Green, South Korean K.J. Choi, and Englishmen Peter Baker and Paul Broadhurst.
Broadhurst lamented a brief lapse in concentration on the seventh hole, resulting in a failed tap-in.
"My balance wasn't quite right, and I sort of pulled out of it," Broadhurst said.
"Putter went past the ball - and went past it. I checked with the referee. I had made a legitimate attempt at it but I sort of went back and didn't get to the ball. So I had to call that."
Scott Hend is two behind the leader at the Senior Open at Carnoustie Golf Links. (Chris Symes/AAP PHOTOS)
Hend (70) is alone in sixth place at 2-under after a spirited surge over his final seven holes.
He chipped in for an eagle on the 12th hole and recorded birdies on Nos.14-15 and 18 to offset a bogey on No.16.
Padraig Harrington, a two-time runner-up who fell to Alex Cejka in the playoff last year, carded a 1-under 71 and sits in a congested tie for seventh place.
"Yeah, OK with it," Harrington said of his round. "I think I played well at times out there, and then I struggled the last couple of holes and made some good saves on 16 and 17. I think 71 is fine. It's well in there."
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