The 40-year-old Rumford, two strokes ahead after round one, signed for a par 72 to be tied for the lead with playing partner and former world number one Westwood on eight under par.
"I'm happy to have signed for 72," Rumford said. "Happy to be tied for the lead and obviously happy that I'm in that top eight.
"You would have to lie to say that you're not aware of your opponent, but I'm not playing against (Westwood), I'm not changing my game plan or swing thoughts or nothing, I'm just playing golf."
Westwood thrived in tough conditions at Lake Karrinyup Country Club by carding a two-under-par 70 to ensure he and Rumford will play together again on Saturday.
Dane Thorbjorn Olesen carded a 67 to be at seven under alongside Briton Grant Forrest, Thailand's Prom Meesawat and Australian Lucas Herbert.
"We both played well," Westwood said. "We haven't given too many shots away, made a few putts and I chipped in a couple of times. It's been fun over the first couple of days.
"I'm just trying to shoot as low as possible. I know if I carry on playing like I'm playing and a few putts start to go in, because I didn't really hole any putts today, I'll be thereabouts on the leaderboard.
"To be perfectly honest, match play is anybody's. You're going to need a lot of luck to win it."
(Reporting by Sudipto Ganguly,; editing by Ed Osmond)
