Green produced nine birdies in an eight-under 63 to sit third, two shots behind leader Bernhard Langer going into the final round at Phoenix Country Club in Arizona.
Looking to cap a wonderful first full season on the senior tour, 53-year-old Green will lift the Charles Schwab Cup for the season-long points race if he wins the tournament title on Sunday.
He boosted his cause by producing the lowest front nine - seven-under 29 - in the tournament's history.
He was at No.6 in the standings going into the event, having posted four runner-up finishes among eight top-10 placings this year, but is still seeking a first win on the tour.
The top 36 players on the points list qualified for the event.
Standing ahead of Green in the battle for the tournament title is German legend Langer, who shot his age - 67 - once again to reach 13 under and put himself in position to extend his PGA Tour Champions winning streak to 18 years.
Langer is out of the points race battle at No.22 in the standings.
Another prolific winner, New Zealand's Steve Alker, is one shot behind Langer in second place after a 63 and well placed to secure the Charles Schwab Cup for a second time from No.2 in the rankings.
"The whole goal was to defend the championship this week and, if I do that, then the Schwab Cup's a bonus," Alker said.
Points race leader Ernie Els posted a 71 to be tied 14th.
The other five Australians in the field were out of contention for the Cup before the event began.
Best-placed was Rod Pampling who fired a 65 to be tied seventh at seven under, with Greg Chalmers (69) and Mark Hensby (73) tied 22nd at one under, followed by Cameron Percy (78, tied 29th, three over) and Stuart Appleby (71, tied 32nd, five over).
Langer, who has shot his age or better 22 times on the tour, has a Champions-record 46 career victories but none so far this year as he has fought back after tearing his left Achilles playing pickelball in February.
"I've got to continue to play aggressive and hit as many fairways as I possibly can and as many greens," Langer said. "The key for me is making putts. I've got to try and hit 15, 16, 17 greens in regulation if I can and make a bunch of putts."
"It would mean a great deal because of what I've been through this year," Langer said. "And also, I didn't win this year. It's the first time in my Champions Tour career.
- with AP
