Mr Dutton, who was immigration minister at the time, has confirmed the 2016 lunch meeting but has denied providing assistance to Huang Xiangmo.
The Chinese billionaire was last year banned from re-entering Australia on ASIO advice he posed a threat of foreign interference because of his links to the Chinese Communist Party.
A joint ABC Four Corners, Sydney Morning Herald and The Age investigation exposed the meeting which was arranged by former Liberal minister turned lobbyist, Santo Santoro.
Mr Turnbull said Prime Minister Scott Morrison could not dismiss the issue as "gossip in the Canberra bubble".
"The buck stops with him. I know what it's like to be prime minister and ultimately, you're responsible. And so Scott Morrison has to deal with this Peter Dutton issue."
The one-on-one meeting followed Mr Dutton's approval of a request to have a private citizenship ceremony for Mr Huang's wife and daughter inside former Labor Senator Sam Dastyari's office in January 2015.
The private event allowed the two to gain their citizenship faster than others who have to wait for a public ceremony.
Mr Dastyari, who was forced to resign from Parliament in 2017 after the extent of his dealings with the influential Chinese businessman were revealed, said he wrote to the immigration minister asking for the expedited citizenship for Mr Huang's family.
"I thought there was a snowfake's chance in hell that this was going to get approved and the pace and speed of which it got approved at the time surprised me, and in hindsight concerns me," Mr Dastyari told the ABC.
Mr Santoro said his services were limited to providing an introduction and denied Mr Huang's citizenship was discussed at the Dutton meeting.
ASIO on threat of foreign interference
The revelations are part of the ABC, The Age and Sydney Morning Herald's joint investigation into the extent of Chinese-backed political activity in Australia.
Australia's spy chief told estimates on Monday the threat posed by foreign interference remains at an "unprecedented level".
Duncan Lewis, the director general of the Australian Security Intelligence Organisation, expressed the sentiment when asked about such interference at a budget estimates hearing in Canberra.
"I have made very plain to this committee on previous occasions that the threat from foreign interference and foreign espionage in Australia was running at, what I described then and I describe again, as an unprecedented level," he said on Monday.
Mr Lewis said he does not respond to media shows or reports and does not single out countries that may pose foreign interference risks.
"We have a challenge here with foreign interference in Australia," he told senators.
"It comes, as I have said on a number of occasions, from a wide range of sources. I have not been country-specific and I will not be country-specific."
But Four Corners has laid bare examples of the Chinese government and its connections attempting to influence Australian political activities.
They include Chinese government officials questioning two Australian citizens about their connections to former China advisor to Malcolm Turnbull, John Garnaut, who worked with ASIO in 2016 on a confidential investigation into Beijing's influence campaign in Australia.
One of the people questioned was now-detained writer Yang Hengjun, who Mr Garnaut says was intercepted by Chinese officials in Sydney in 2018.
He was later detained in January by Chinese officials on suspicion of endangering China's national security after flying from the United States to Guanghzou. His family have not been able to see him since.
The Four Corners report said the Chinese consulate also pressed Sydney's Georges River Council not to allow a Chinese newspaper, the Vision Times, to support a Chinese New Year event.
And it has pointed to figures allegedly linked to the Chinese government who have been prominent political donors and fundraisers, such as Melbourne millionaire Tommy Jiang.
Legislation banning foreign donations cleared federal parliament in November.