Westwood joins Rumford at the top of Perth's World Super 6
Westwood shot a two-under 70 in tricky winds on Friday to move to eight-under overall, alongside Rumford.
Perfect conditions greeted the field in round one but blustery winds on Friday meant it was much tougher to make a dent on the scorecard.
Rumford shot an eight-under 64 in round one but had to settle for an even-par 72 on Friday.
Westwood was on track to take the overall lead heading into round three before his short putt on the 18th hole rolled just wide.
Amateur Min Woo Lee is at five-under.
Thailand's Kiradech Aphibarnrat (five-under) briefly shot to the lead on Friday before four-straight bogeys on his back nine saw him drop back to the pack.
England's 2016 Masters champion Danny Willett missed the cut after finishing a disastrous seven-over, while countryman Andrew 'Beef' Johnston (five-over) will also miss the weekend.
Rumford, a hometown hero, said the conditions on Friday were as tough as he had ever seen them at Lake Karrinyup.
"It was right from the word go. It just hit you on the first tee. It was 'bang' straight in the face," Rumford said.
"Every single hole looked differently to yesterday. You couldn't really go out and chase it today."
Only the top 24 players will qualify for the final-round match play component, when golfers will go head-to-head in a series of six-hole shootouts.
The top eight players entering the final day earn a crucial bye in the first round of match play.
Westwood said his main priority is to finish in the top 24, rather than put pressure on himself to get a top-eight seeding.
And he said the match play component made it hard to predict who will come out on top.
"You're going to need a lot of luck," he said.
"I think it's exciting for the fans and the format of the game. Golf needs something like that.
"It's very do or die over the first couple of holes of match play."