The prime minister has left the door open to a potential referendum on freedom of movement with the European Union.
Asked by broadcasters if he could introduce free movement without a public vote Andy Burnham said "no", but sources told the BBC no decision on that has been made.
The PM added that he would "never move ahead of the British public when it comes to Brexit".
Burnham has ruled out major changes to the UK's relationship with the EU before the next general election but says he wants start a conversation about potential options, with these expected to be set out around the time of the EU summit next month.
Freedom of movement, which allows EU citizens to travel, live, study and work in any member country, is a precondition of membership of the EU's single market.
The single market enables goods, services and people to move freely between member states, with countries also applying many common rules and standards.
The prime minister is thought to be against the idea of leaping quickly to a return to freedom of movement.
Sources indicated no firm decision had been made on whether free movement would need to be rubber-stamped by a referendum.
Burnham is said to believe that a conversation about the UK's future relationship with Brussels needs to happen first.
Migration dominated debate during the 2016 Brexit campaign, with the Leave campaign raising issues about freedom of movement allowing any EU citizen to move to the UK.
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.