The industry has been in the spotlight after three dogs died in a week.
Racing was also shifted from Manukau Stadium to Cambridge last week after a "concerning increase in injuries".
Luxon had during a televised election debate voiced support for a ban, and reiterated that after the Cabinet meeting on Monday saying he was personally "aligned with where the previous government was heading" but a ban would be "something for the Racing Minister to consider".
The Post on Wednesday reported Racing Minister Winston Peters' comments he was having "a lot of work" done on the industry. He said they were "seriously looking at all aspects of it" and expected to have an outcome in three weeks but - asked if Luxon was premature to talk of a ban - said "he's got a minister now who knows more about it than he does".
Labour's animal welfare spokesperson Rachel Boyack challenged the prime minister to urgently ban greyhound racing.
"The government will have Labour's full support if it proceeds with a ban on greyhound racing," she said.
"All of the work was happening so that a decision could be taken by Cabinet after the election. The govenrment has now completed its 100 day plan, so it is time now to get on and ban this industry."
"It's time for the ban and we've had the prime minister recently state that he personally supports it, and so they can act and they should act."
She pointed to the review commissioned by the party's former Sports minister Grant Robertson, which in 2021 found the industry had failed to improve animal welfare systems to avoid injury or death and maintained a "culture of silence" against reform of the sport.
Robertson at the time said the industry was "formally on notice" and gave the industry an ultimatum to address concerns by the end of 2022, but Labour's Racing Minister Kieran McAnulty in October that year said he was waiting on a final report from the Racing Integrity Board.
In mid-2023 McAnulty - who was named the Cyclone Recovery Minister after a string of ministerial resignations - said the report's release was delayed by his other responsibilities and he likely would not have time to bring a greyhound ban discussion to Cabinet before the election.
Boyack rejected suggestions Labour's internal ructions were what had prevented a ban being in place by now.
"The entire Cabinet was focused on the cyclone recovery, it was a massive issue that affected our whole team, it was the largest natural disaster of its kind in New Zealand so I just don't accept that," she said.
"It's a big decision to ban an entire industry, it's not one you can just take lightly. You do need to give all stakeholders and the industry the opportunity to engage in that process, to fix the issues."
She argued the industry had been given that opportunity, but "they have not acted and it's time for the government to act".
"One of the big issues was around building straight tracks: we know greyhounds get injured when they go around curved tracks and they haven't built any straight tracks - despite promising the select committee I sat on that they were going to build one imminently. It hasn't happened."
She defended other industries when asked about the value of activities like rodeo, however.
"I want to be really clear that we're talking about one industry. The horse-racing industry is one of our strongest export markets, we have a fantastic industry that operates safely.
"The horsing industry does not have anywhere near the same number of animal welfare issues we have around greyhounds ... I don't want to bring that industry into disrepute, we have some of the best horse breeders and trainers in the world."
She said the horse racing industry had engaged with animal welfare organisations - unlike the greyhound racing industry.
RNZ
