The minister concedes he knew the man when they worked together in the Queensland Police, but said he had not spoken to him in 20 years.
The case concerns one of two au pairs Home Affairs minister Peter Dutton spared from deportation in 2015.
The young Italian woman arrived in June of that year and had her visa cancelled by border officials, who believed she intended to work on a tourist visa, but was later spared from deportation by a ministerial intervention from Mr Dutton’s office.
Leaked emails have already revealed the request came from Russell Keag, a former colleague of the minister.
The minister and the new Morrison government have downplayed the link, while Labor and the Greens are claiming Mr Dutton misled parliament when he claimed to have no personal connection to the families in question.
Greens MP Adam Bandt has seized on the report and the “huge” allegations.
“Dutton misled [parliament] and must go,” he tweeted.
Labor's shadow immigration minister Shayne Neumann said the prime minister should ask Mr Dutton to explain the link.
"Peter Dutton obviously knew what he was doing and Peter Dutton obviously remained having a personal connection with the police officer with whom he served with in the Queensland Police," Mr Neumann said.
Earlier on Thursday morning, before reports of Mr Quaedvlieg’s letter surfaced in the media, Mr Dutton told Radio 2GB there was a “former senior ABF officer who leaks this information out”.
He did not name the person he suspected of leaking.
“Good luck to him if that’s what he wants to do. He’s obviously very close to the Labor party ... all of that will come out at some point,” Mr Dutton said.
On the weekend, prime minister Scott Morrison defended his cabinet colleague.
“How many people did you work with 20 years ago?” Mr Morrison asked reporters.
Mr Dutton’s department previously spent $10,000 trying to prevent the release of documents about the au pairs in a Freedom of Information dispute with a journalist.