The timing is significant. Rather than simply arguing for closer cooperation with Brussels, the London Mayor has called for the groundwork for rejoining the European Union to begin now. His argument is that Labour should not wait until the next general election before presenting voters with a fully developed alternative.
That immediately creates a political dilemma for Mr Burnham.
The Prime Minister has previously insisted that Labour’s existing Brexit commitments remain in place. The party’s 2024 manifesto ruled out returning to the single market, the customs union or freedom of movement. Those commitments were designed to reassure voters that Labour was not planning to reverse Brexit.
But the political environment has changed considerably since then.
Polling, economic concerns and continuing arguments about Britain’s relationship with Europe have encouraged some senior Labour figures to question whether the party should eventually go further. Sir Sadiq’s comments represent one of the clearest calls yet for a future Labour government to consider full EU membership.
For Brexiteers, however, the proposal is viewed as an attempt to reopen a question that was settled by the 2016 referendum.
The referendum produced a clear decision for Britain to leave the European Union, and the process was subsequently implemented by successive governments. Supporters of Brexit argue that reversing that decision would require another explicit mandate from voters rather than being introduced indirectly through a general election manifesto.
This is why Mr Rosindell’s response was so forceful.
The Reform UK MP accused Labour politicians of refusing to accept the referendum result and warned against what he described as an attempt by the political establishment to take Britain back into the EU.
That argument is likely to become increasingly prominent if Labour begins openly discussing membership before the next election.
The debate is also exposing divisions inside the Labour movement itself.
Sharon Graham’s intervention is particularly significant because Unite has historically been one of the most influential organisations within Labour politics. While she supported closer cooperation with European countries, she rejected the idea of simply attempting to undo Brexit.
Her argument was that Britain should focus on using the powers it has outside the EU rather than constantly debating whether to reverse the referendum.
That position could resonate with Labour voters who supported Brexit but remain sympathetic to the party.
The political calculation for Mr Burnham is therefore complicated.
A pledge to rejoin the EU could appeal to younger voters, many of whom were more likely to support Remain in 2016. It could also strengthen Labour’s appeal among metropolitan voters and those who believe Brexit has increased trade barriers and reduced Britain’s economic opportunities.
But it could simultaneously alienate Leave voters who helped Labour make gains in traditionally Conservative constituencies.
That electoral coalition may be especially important for a Labour government trying to maintain support outside London and other major cities.
The question is therefore not simply whether rejoining the EU would have economic advantages or disadvantages. It is whether such a commitment could be turned into a winning political proposition.
Brexit supporters argue that the country should concentrate on improving its existing relationship with Europe rather than attempting to reverse membership altogether.
There are already areas where Britain and the EU can cooperate more closely without restoring full membership. Defence, security, scientific research, trade, energy and cross-border crime are all areas where practical cooperation could potentially be expanded.
That middle ground could offer Mr Burnham a way of managing pressure from both sides.
However, Sir Sadiq appears to want the conversation to move much further.
His reference to “pitch rolling” suggests a long-term political campaign rather than an immediate policy announcement. The argument is essentially that Labour should begin persuading voters now that EU membership is desirable, allowing a future manifesto to contain an explicit commitment without appearing to have been produced suddenly during an election campaign.
That could transform Brexit into a major issue at the next election.
For Conservatives and Reform UK, such a development would provide an obvious political target.
They could portray Labour as attempting to overturn the referendum and accuse Mr Burnham of preparing to surrender powers over trade, immigration and regulation to Brussels.
Reform UK in particular would be expected to make Brexit central to its campaign, presenting itself as the party committed to maintaining Britain’s independence.
The Conservatives would face a more difficult calculation.
The party governed Britain through much of the Brexit process, but its support for Brexit has not always translated into a unified position on Britain’s future relationship with the EU. Some Conservatives favour closer economic ties while others argue that Brexit should be defended more aggressively.
Labour’s internal debate could therefore create opportunities for both opposition parties.
For Mr Burnham, the most immediate challenge is to avoid allowing the issue to overshadow his wider domestic agenda.
The Prime Minister has sought to present himself as focused on government delivery, including the economy, public services and household finances. A prolonged internal Labour argument about EU membership risks shifting attention away from those priorities.
It could also reopen divisions within Labour at precisely the moment when the party needs to demonstrate unity.
The comments from Sir Sadiq are particularly awkward because they come from a senior Labour figure rather than an opposition politician. If more MPs, ministers or party activists begin publicly supporting rejoining the EU, the Prime Minister could find himself repeatedly forced to explain whether he agrees.
Every attempt to keep the issue at arm’s length could then be interpreted as evasive.
Every indication that the door is open could be seized upon by opponents as evidence that Labour is preparing to reverse Brexit.
There is also the question of what “rejoining” would actually mean.
Returning to the EU would not necessarily mean simply restoring the situation that existed before 2016. The European Union has changed, Britain has changed and the terms of any future membership would have to be negotiated.
Issues such as the single market, customs union, freedom of movement, the euro and other EU obligations would all have to be addressed.
This is one reason why critics argue that politicians should be cautious about presenting EU membership as a simple solution to Britain’s economic problems.
Even supporters of closer European integration acknowledge that rejoining would be a lengthy political and diplomatic process.
Sir Tony Blair’s recent intervention has already pushed the issue higher up the political agenda. Sir Richard Branson’s criticism of Brexit has added another prominent voice from the business world.
Sir Sadiq’s demand takes the debate another step forward by asking Labour to prepare for rejoining rather than merely improving relations.
Whether Mr Burnham accepts that challenge remains unclear.
For now, the Prime Minister appears to be balancing two competing pressures: Labour figures who want a much closer relationship with Europe and voters who do not want Brexit reversed.
His decision could have consequences well beyond the next manifesto.
If Labour eventually commits to EU membership, the next general election could become partly a referendum on Brexit all over again. If it rejects the idea, some pro-European voters may accuse the party of being too cautious.
Either way, Sir Sadiq has succeeded in forcing a question back into British politics that many Westminster politicians had hoped would remain dormant.
Nearly a decade after the referendum, Brexit remains unresolved as a political argument even if the legal process of leaving the EU has already happened.
The battle now is increasingly about what comes next.
For Andy Burnham, that means deciding whether Labour should defend the Brexit settlement, seek a closer relationship with Europe, or ultimately prepare the country for another historic change.
And with senior Labour figures already demanding that the groundwork begin, the Prime Minister may find it increasingly difficult to keep that decision off the political agenda.