Asked earlier on BBC Radio 4’s Today if he wanted the UK to rejoin the EU, Burnham replied that he wanted to “look at the options”.
The PM added that he had previously advocated such a move within his lifetime, whilst acknowledging it was “not necessarily the case that that is now”.
French President Emmanuel Macron said it would be “very good news” if the UK wanted to rejoin the EU, while Spain’s Prime Minister Pedro Sanchez said his country would welcome Britain back “with open arms”.
Lib Dem leader Sir Ed Davey said his party had put forward “concrete plans to boost our economy, strengthen our defences and cut people’s taxes by joining the single market and a customs union”.
He added: “Talk is cheap. Burnham must stop with all the ‘maybes’ and get negotiating that better deal for Britain today.”
The party’s Europe spokesperson Al Pinkerton said last week there is “no rejoin on the table”, adding Britain would effectively be treated as a “fresh entrant state” if it wished to join the bloc again.
Reform UK deputy leader Richard Tice accused Burnham of “trying to sneak us back into Brussels via the back door”.
Tice said: “He says Brexit has done more harm than good but what’s done real harm is a Labour government that can’t deliver growth, can’t control migration and now wants to outsource the job to Brussels.”
He added Reform would “never let this betrayal go unchallenged”.
In his Labour conference speech on Tuesday, Burnham blamed Brexit for a loss of control over both immigration and the economy, telling delegates he did not believe the current relationship was “good enough”.
He told activists he wanted to use the rescheduled summit later this year to lay out “different options” for the UK’s long-term relationship with Europe.
He promised to “find consensus” on an option to clarify the UK’s position, although did not offer further detail of how this would be measured.
In response to Burnham’s conference speech, Conservative deputy leader Alex Burghart said: “Labour cannot pin the blame for their disastrous economic decisions on a vote that happened over a decade ago.
“Besides, they are already dragging us into the European Union via the back door with their EU reset.”
Green Party leader Zack Polanski said: “All Andy Burnham offers on repairing our relationship with our biggest trading partner is a promise to keep our options open.
“The Green Party is very clear that we should rejoin the customs union to start reversing the damage, and full membership of the EU remains the best option for the UK.”