The prime minister says he would welcome an "honest discussion" about the prospect of returning to free movement between the UK and European Union.
Speaking to the Sunday Times, Andy Burnham said he wants pursue a new type of relationship with Brussels, potentially including full EU membership or joining the single market, which would require free movement.
"I think you have to have a very factual, quite evidence-based debate about this," he told the newspaper.
Shadow home secretary Chris Philp said there could be "no return" to free movement after the 2016 EU referendum, while Reform UK leader Nigel Farage said the prime minister's comments were a betrayal of voters who supported Brexit.
Burnham has previously committed to Labour's 2024 election manifesto, which ruled out the return of free movement, single market membership or a customs union.
He has said that any fundamental changes would require a mandate at a general election, which is due by 2029.
Migration dominated debate during the 2016 Brexit campaign, with the "leave" campaign raising issues about rules allowing any EU citizen to move to the UK.
But the prime minister suggested the type of migration experienced during the years of EU membership was "different" from the challenges faced now.
"It was generally younger people, not always, but spending a few years, then going back," he said in his told the Sunday Times., external
"And obviously the ease of that worked for British people as well."
He acknowledged that freedom of movement "had very significant challenges associated with it".
"Revolving door migration where people would come and then go back is different," he said.
"The balance of migration has changed, hasn't it, since then? I think we've just got to have an honest discussion about those things."
Burnham's willingness to re-open some of the most sensitive debates in British politics has been a feature of his first months in power.
In an interview with the BBC in September, Burnham said Britain could go "all the way" towards rejoining the European Union.
However, in his latest remarks, the prime minister stressed: "I can't see myself ever recommending anything that would involve joining the euro or Schengen."
The Schengen Agreement allows passport-free travel across EU countries. While the UK did not previously abandon the pound while it was an EU member, the bloc has pushed new members to adopt the single currency.
The remarks were made ahead of his meeting with EU chiefs at a summit on 20 November in Brussels, where he is expected to pitch options for a reset of relations with the bloc.
The shadow home secretary said EU free movement would amount to "handing back control of our borders and allowing uncontrolled migration from across Europe".
Philp said: "The British people voted to end free movement, and there can be no return to it."
Burnham won the Greater Manchester seat of Makerfield at a by-election in June, ahead of him becoming prime minister the following month. The constituency voted to leave the European Union in 2016.
As a candidate in the by-election, Burnham had promised not to "re-run" Brexit arguments and said he was "not proposing that the UK considers rejoining the EU".
In response to the Sunday Times interview, Farage said: "This is a total betrayal of Brexit Britain and his Makerfield pitch".
"He is also deluded to say immigration was different when we were members of the EU. We will fight this dishonesty," the Reform UK leader added.