Speaking to Radio 4’s Today programme, Burnham said he did not want to leave the question of the EU relationship “hanging” and he wanted to be clear on “where we want Britain to go in the next decade”.
Asked whether he wanted the UK to rejoin the EU, he replied that he wanted to “look at the options”.
He added he had previously advocated such a move within his lifetime, whilst acknowledging it was “not necessarily the case that that is now”.
“We could stay as we are. That’s definitely an option, if people think this is the right place to stay,” he told the programme.
“We could look at what [former Chancellor] George Osborne has said about a customs union, we could look at what the Liberal Democrats said at their conference about the single market, or we could go all the way.”
“So there are clearly options that need to be considered, and we have to look at what is doable and what would be the pros and cons of each,” he added.
Jukka Salovaara, the EU’s ambassador to the UK, said last week: “The doors are open but it’s very much up to the UK public to identify and decide on its level of ambition when it comes to deepening cooperation with the EU.”
Lib Dem leader Sir Ed Davey said his party has 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.
Pinkerton also said the British public need to be “with you in a sustainable way” and the UK would have to show to its European partners that “it’s in their advantage too that Britain re-engages”.
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 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.”