The Canadian Prime Minister said on Friday that France was one of the few countries with the ability to read the flight and cockpit data recorders from the jet, which he said were badly damaged.
It comes as Mr Trudeau said his government will provide financial support to the families of the 57 citizens and 29 permanent residents of Canada who died in the crash.
He said the families will receive 25,000 Canadian dollars (£14,600).
Mr Trudeau said he still expects Iran to compensate the families but said they need help now for funerals, travel to Iran and bills.
Iran says it shot down Ukrainian International Airlines flight 752 last week by accident, killing all 176 people aboard.
"Iran does not have the level of technical expertise and mostly the equipment necessary to be able to analyze these damaged black boxes quickly," Mr Trudeau said.
"There is a beginning of a consensus that [France] would be the right place to send those black boxes to get proper information from them in a rapid way and that is what we are encouraging the Iranian authorities to agree to."
In the week since Tehran announced its military had shot down the aircraft, Iran's government has said international authorities looking into the tragedy would have access to the black-box data, but that has yet to happen, slowing the investigation.
More than two thirds of Canadians are not confident there will be a full and accurate account of the disaster, an Angus Reid Institute poll released on Friday said.
Iranian Foreign Minister Mohammad Javad Zarif held a rare face-to-face meeting with his Canadian counterpart, Francois-Philippe Champagne, on Friday in Muscat, Oman. The two countries have not had diplomatic relations since 2012.
In a statement, Canada's foreign ministry said Mr Zarif agreed on the need for "a transparent analysis of the black box data," and that the ministers "discussed the duty Iran has towards the families of the victims – including compensation".
Mr Zarif wrote on Twitter after the meeting that both countries' experts would continue to exchange information, adding "politicisation" of the tragedy must be rejected and that the focus should be on the victims' families.
Mr Trudeau said about 20 families of Canadian victims had requested the return of the bodies and that he expected the first remains to be repatriated in the coming days. He gave no further details.
The bodies of all 11 Ukrainians who died in the crash have been identified and will be transported to Ukraine on January 19, the interior ministry in Kiev said in a statement.
With Iran facing a deepening crisis as it grapples with unrest at home and rising pressure from abroad, the Islamic Republic's supreme leader Ayatollah Ali Khamenei delivered his first Friday prayers sermon in eight years.
He described the shooting down of the plane as a "bitter tragedy that burned through our heart".
But he stopped short of offering a direct apology for the disaster and instead urged Iranians to unite and show solidarity by turning out in numbers for the February parliamentary election.