However, whether he will be able to weave a piece of Jacinda Ardern-style magic and get a concession from Anthony Albanese appears extremely doubtful.
The issue represents a test for Mr Luxon in how far he is able to influence the bigger partner in the trans-Tasman alliance.
Mr Luxon will spend Thursday in Sydney before heading to the capital on Friday for the annual Australia New Zealand Leaders Meeting.
The two countries are steadfast allies, with a wide government-to-government work plan that could rival any two nations on Earth.
Since coming to office last year, Mr Luxon's government has signalled a closer alignment on defence and foreign policy, and further work on decarbonisation.
Australia's practice of deporting Kiwi criminals who have grown up in Australia remains the biggest rock in the relationship's shoe.
In the past decade, Australian ministers have sent more than 3000 criminals to New Zealand using powers under section 501 of the Migration Act.
New Zealand recognises Australia's ability to do so, but has consistently protested against deportations of Kiwis-in-name-only; those who have spent long stretches.
"Do not deport your people and your problems," Ms Ardern memorably told Scott Morrison in 2020, though her arguments fell on deaf ears.
That was, until Mr Albanese's government came to office in 2022, when the NZ Labour leader was able to convince the new Australian prime minister to soften the policy.
"Just after Albanese became the Prime Minister, she came straight over and had dinner with him at Kirribilli House," Dame Annette King, who was then the High Commissioner to Australia, told AAP last year.
"He reaffirmed to her what he was going to do. He was incredibly enamoured with her advocacy."
In early 2023, Mr Albanese announced the government would give more consideration to the length of residency in Australia and the family impact before deporting Kiwis.
However, in June, Australia back-flipped and issued a new directive to under pressure when it became known that individuals who had escaped deportation had committed new crimes in Australia.
Mr Luxon has said he "regrets" Australia's move, which is "just not right".
Speaking in Wellington on Wednesday, Mr Luxon said he'd raise it in bilateral talks "pretty directly".
Janet Wilson, who was press secretary for Mr Luxon's predecessor as National party leader, said Kiwis would be watching to see how he tackled the issue.
"The 501s will be his test in the next two or three days, and how strong his language is going to be," she told AAP.
Compounding his political problem, Mr Luxon's centre-right National party has pledged a crackdown on gangs - but membership of gangs have soared in recent years due to 501 deportees who arrive without family networks in New Zealand.
"They are actually causing society, New Zealand society, irreparable harm," Ms Wilson said.
"It's somewhat ironic that the New Zealand political party that has talked tough and hard when it comes to dealing with crims is sounding comparatively weak when it comes to the 501s."
Before arriving in Canberra, Mr Luxon will meet NSW Premier Chris Minns on an infrastructure-focused day in Sydney.
He will also give a foreign policy address at the Lowy Institute.
- By Ben McKay in Wellington
