Burnham has said he is committed to Labour’s 2024 election manifesto, which ruled out a return of free movement, single market membership or a customs union.
The PM has stressed any changes would require a mandate at a general election, which is due by 2029.
However, he has suggested potential options for the future could range from no changes to the UK-EU relationship, to rejoining the EU customs union, the single market or a new in-out referendum on full EU membership.
In an interview with the Sunday Times, external, Burnham said he would welcome an “honest discussion” about the prospect of returning to free movement between the UK and the EU.
The PM argued that the type of migration had changed since Brexit, with more foreigners moving permanently.
In contrast, he suggested that when the UK was a member of the EU, migration was “generally younger people”, who would spend a few years in the country before returning home.
However, Burnham added: “I can’t see myself ever recommending anything that would involve joining the euro or Schengen.”
The Schengen Agreement, which the UK was not part of when it was in the EU, allows passport-free travel across member countries.
The UK was able to keep the pound when it was an EU member, but the bloc has pushed new members to adopt the Euro.
Burnham has argued that since Brexit, the UK has lost immigration control the country previously had.
“For instance, the small boats crisis has arisen in the last 10 years because we lost that returns agreement when people came here that we could work with other countries to return them,” he told broadcasters.
The Conservatives and Reform UK both oppose a return to free movement with the EU.
Conservative leader Kemi Badenoch argued most people who voted for Brexit did so because they did not want free movement, while Reform leader Nigel Farage said such a move would be “a total betrayal of Brexit Britain”.
However, the Liberal Democrats have argued that allowing freedom of movement would help economic growth.