The prime minister is expected to tell the Commons later it would be a "dereliction of duty" to turn down a closer relationship with the European Union - if one can be negotiated with Brussels.
In a statement to MPs on Monday, Andy Burnham will argue it is time to "face up to the reality that Brexit has done more harm than good" and "the status quo is not good enough for Britain."
He first pledged to explore closer ties with the bloc at a UK-EU summit later this year at the Labour Party conference last month.
Addressing the Commons, Burnham is expected to say that a new deal with the EU could help tackle illegal immigration and boost economic growth.
He will claim that if he could secure that then "the vast majority of the public would expect us to take it".
Last month, Burnham told the BBC it was "possible" a future Labour election manifesto could promise a new in-out referendum on the EU.
A summit between the UK and the EU was due to take place in the summer but was postponed after Sir Keir Starmer announced his resignation. It has been rescheduled for next month.
The prime minister hopes to secure tweaks to the current arrangements between London and Brussels and also begin a wider conversation about the long term relationship between the UK and its nearest neighbours.
A No 10 source said: "Andy understands why some politicians don't want to face up to the reality that Brexit has not delivered what they once promised.
"But it is time for a different conversation about what our long-term relationship should be. Sticking with the status quo and not working relentlessly to secure a good deal that boosts growth would be a major dereliction of duty."
In his speech at Labour's conference, he said "immigration control" had been "weakened by Brexit" because the UK "lost collaboration with our European partners. And we are still having to work hard to bring it back".
Speaking at her party conference last week, the Conservative leader Kemi Badenoch said it was "a terrible idea" to "reopen the Brexit debate."
"Can you imagine having to go through that again? So much time lost, arguing with ourselves with the EU. Going round in circles, businesses not knowing whether they should invest. He wants to do it all again."
The Commons sits on Monday for the first time since 15 September, having been in recess during some of the party conferences.