Malcolm Turnbull has attempted to clean up a public difference of opinion between himself and Peter Dutton, acknowledging that ministers did discuss the composition of Australia’s migration program.

The spat was triggered by a report in the Australian on Tuesday saying the home affairs minister had previously suggested capping Australia’s immigration intake at 170,000 a year – which would have meant a cut of 20,000 people last year – but the proposal was quashed by the prime minister and the treasurer, Scott Morrison, before it made it to cabinet.

Turnbull subsequently declared the report “false” and “completely untrue”. Dutton then contradicted the prime minister, essentially confirming that he had discussed the issue with colleagues.

On Thursday morning Turnbull attempted to argue during an interview on 3AW that the initial report had referenced a cabinet deliberation, which it hadn’t.

“What was initially said in media, I think in the Australian, that there had been a submission brought to cabinet by Peter Dutton to reduce the ceiling of permanent migration ... and that he had been rolled by me and Scott Morrison,” he said.

He said the issue had not gone to cabinet, but “if you are asking me, do ministers discuss migration and migration levels and the composition of the migration program – well of course we do.

“It would be strange if we didn’t. And I might say the permanent migration ceiling, which has been set at 190,000 for some time, and which we were well below last year and we expect to be below this year, that is reviewed every year.”

The row is connected to a bout of positioning connected to the government’s loss of its 30th Newspoll this week, a metric which assumed oversized significance because Turnbull had invoked it as one of the rationales to dump Tony Abbott from the Liberal leadership in 2015.

The government attempted to respond to the loss with a full court press of the cabinet, with all the key senior figures deployed in media interviews to decree the prime minister’s leadership was safe.

As the key players expressed loyalty, two conservatives also articulated their desire to lead the Liberal party if the top job became vacant – Dutton and Morrison.

Some government figures believe Tuesday’s story in the the Australian is connected to leadership positioning by Dutton, but others believe it is connected to a broader fight within the government about the policy direction required to turn around the government’s stubbornly negative poll trend, or to save the furniture.

Some government conservatives believe cutting the migration rate would be popular in the regions and in outer metropolitan areas, delivering the Coalition a poll bounce, because voters associate high rates of migration with congestion in the major cities, and suboptimal infrastructure and service delivery.

Turnbull and Morrison have resisted the arguments which have been articulated publicly by Abbott and in a more muted fashion by Dutton. During a separate interview on 2GB on Thursday morning, another senior conservative, the finance minister, Mathias Cormann, said there had been “no suggestion that we should be reducing the overall net permanent migration further”.

On 3AW, Turnbull was asked whether there was a case to freeze the current rate. The prime minister said the trends driving an increase in foreigners in Australia were temporary modes of entry: students, tourists and business visitors.

Turnbull said there are 200,000 more foreign students in Australia than there were a few years ago.

Implicitly referencing the internal view points, Turnbull said: “If you feel there are more foreigners on the tram and you can’t get a seat on the tram, if that’s your perception, they are most likely students and visitors.”

He repeatedly ducked questions about the desirable rate of population, arguing it wasn’t about a number, it was about maintaining the level of population consistent with a decent standard of living.

He said the government was focused on city planning to ensure liveability. He was planning infrastructure projects “in a way no previous prime minister has done”.

Alan Tudge, the citizenship minister, said the migration rate would be the same this year: “I think the important thing about this debate about population policy is we need to make sure the infrastructure is there to cater for population growth.

“It’s why it’s so pleasing to see the prime minister announce the Melbourne airport rail link. It’s a link we have wanted for decades. That will take a huge amount of congestion off the Tullamarine freeway.”