“To the best of my knowledge, I have not socialised with, met with or had personal contact with the man involved,” Mr Dutton told parliament on Monday.
“He does not have my personal phone number or my personal email address.”
But, in an email Mr Dutton tabled in Parliament late on Monday afternoon, a June 2015 email from Mr Keag indicates a familiarity between the two.
"Peter. Long-time between calls," the email reads.
"I need advice on a matter that has occurred today. An Italian student, Michela Marchisio is being held after her visa has been cancelled."
Mr Dutton’s relationship with the man has been the subject of intense focus in recent weeks over a case where a young Italian woman was detained at Brisbane international airport in 2015.
Border Force officials suspected she was planning to work for Mr Keag as an au pair despite only holding a tourist visa and planned to deport her, but she was later spared by a ministerial intervention from Mr Dutton.
Labor and the Greens claim the fact Mr Dutton knew the man when they worked together in the Queensland police means the minister misled the parliament when he said he had no personal connection to the families involved.
They accuse the powerful Home Affairs minister, who unsuccessfully challenged for the Liberal leadership last month, of misleading the parliament.
But Mr Dutton says the relationship was more distant than his political opponents make out.
“The use of terms such as ‘personal connection’ … signify a much closer relationship between two people than working in the same organisation two decades ago and not speaking to each other since that time,” he said.
Greens MP Adam Bandt, who originally asked the question about a personal relationship, said he would move a no-confidence motion against Mr Dutton this week.
“If Scott Morrison won’t stand him down then the parliament should,” Mr Bandt told reporters.
Labor frontbencher Tony Burke said Mr Dutton had broken the rules and should stand down.
“The one rule that a minister has to keep is you don't mislead the parliament, and on the facts that we've got available at the moment, Peter Dutton clearly did.”
The Australian published a story based on a tip from a “government source” on Monday that highlighted a case where Italian soccer star Alessandro Del Piero’s nanny was allowed into the country on a tourist visa under the previous Labor government.
But unlike the Dutton cases, the nanny did not get a ministerial intervention from Labor’s immigration ministers Chris Bowen and Tony Burke.
The department approved the tourist visas itself, although the unnamed source claimed the immigration department was “heavied” into granting them by the minister’s office.
Tony Burke said the drop was likely a leak from Mr Dutton’s office. The minister has previously warned he had records of "quirky cases" of visa interventions requested by Labor MPs.
“He’s throwing mud into a gale force wind and it’s landing all over him,” Mr Burke said.