The Australian is three under for the tournament, three behind tournament leader K.J Choi.
Green, who opened with a 69 and was one off the first-round pace set by Canadian Stephen Ames, had to work hard on Friday for his even-par round.
After sinking birdie putts on the sixth and eighth, Green endured a horror patch when he made four bogeys in a row beginning on No.8.
But the left-hander showed great resilience by bouncing back with an eagle on the 12th and a birdie on the 14th to balance his ledger and join Irishman Padraig Harrington (70) in a tie for third.
K.J. Choi leads the Senior Open by two from Canadian Stephen Ames. (Jeremy Ng/AAP PHOTOS)
Choi (69) is the player to catch as the South Korean surged ahead after he eagled the par-5 12th hole.
He matched his opening 69 to open up a two-shot lead over Ames (72) .
"Today's iron shots were better than yesterday and still inside six, seven foot; the putter, reading is very difficult," Choi said.
"Couple of mistakes on some shots and couple of mistakes with the putter but still scored the same as yesterday."
Overnight leader Ames is Choi's nearest chaser going into the weekend. The naturalised Canadian citizen from Trinidad shot par for the day.
Harrington is well placed to become the fifth man to win the British Open and Senior British Open.
"We've only played 36 holes. Feels like we've done 72," Harrington said.
"Been a tough 36 holes with that weather and wind and you've got to all be on, and I'm sure it's going to be another 36 of tough mental fortitude to come."
Scott Hend shot a five-over-par 77 to slide down the leaderboard and join countryman Greg Chalmers (75) at three over for the tournament.
The Australians are in a tie for 21st.
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