Australia are seven wickets away from retaining the Trans-Tasman Trophy and winning the first Test against New Zealand, with Nathan Lyon at the centre of their toil.

Spinner Lyon made an innings-high 41 as the tourists slumped to 164 in their second dig; setting the Black Caps an improbable target of 369 for victory.

A session into their chase, New Zealand were 3-111 at stumps on day three, still 258 runs from a first home victory over their arch-rivals in 31 years.

While a huge task remains, the Kiwis enjoyed their best day of the Basin Reserve Test on Saturday.

The man behind their best bowling effort was also their best performer with the bat, Glenn Phillips.

Phillips (5-45) ripped through the Australian middle order, removing Travis Head (29) and Mitch Marsh (golden duck) in consecutive balls.

Alex Carey survived the hat-trick attempt, but Phillips soon claimed the wicketkeeper’s scalp, adding him to Usman Khawaja (28) and Cam Green (34) on his list of victims.

Not bad for someone who started his international career as a T20 wicketkeeper, and who captain Tim Southee had overlooked for a spell while Australia tallied 383 in their first innings.

“Credit to Glenn, I thought he bowled really well,” Lyon said. “Stuck to his pretty simple game plan … he was too good.”

Matt Henry followed up his first-innings 5-70 with another solid 3-36.

Australia’s limp second-innings effort didn’t even match Cam Green’s unbeaten first-innings 174.

Only Lyon, given his batting average of 12, could stand tall, though he fell short of a career ambition.

The 36-year-old struck three consecutive boundaries off Southee on the way to his second-highest score in 162 innings, but not the Test half-century he has long craved.

“That’s the second time in my career I’ve top-scored, and let’s hope it’s the last,” Lyon said.

It would have been an even better day for New Zealand if they had held their catches.

As it did in the T20 series, poor fielding hurt the Black Caps, with Scott Kuggeleijn, Southee (twice) and Henry Nicholls all dropping regulation chances.

When New Zealand took to the crease just before tea on Saturday, Aussie captain Pat Cummins wasted little time bringing Lyon into the attack.

Lyon (2-27) had Tom Latham caught behind cutting off just his second ball, and followed up with the prize wicket of Kane Williamson after the break.

The former captain glanced a Lyon delivery down the leg side straight to Steve Smith at an unorthodox leg slip for nine.

Smith then pulled off a one-handed stunner at slip to dismiss Will Young off Head’s spin.

From that point, New Zealand stood strong, Rachin Ravindra (56) and Daryl Mitchell (11) putting together a half-century stand to keep Kiwi dreams alive heading into day four.

It remains to be seen whether paceman Will O’Rourke is able to bat, given the 22-year-old hobbled off the field with a tight hamstring in his eighth over.

Success for New Zealand would break multiple records.

Reaching 369 would be their highest successful fourth-innings chase in their 94-year Test history, and be the best fourth-innings chase at the Basin Reserve by almost 100 runs.

It would also end the Black Caps’ 31-year run without a home win against Australia, and keep alive their Trans-Tasman Trophy hopes.