In a Commons address on Monday, the Prime Minister is expected to reiterate his argument from the recent Labour Party conference that Brexit has done "more harm than good".
He will call on all parties to support the Government's approach in seeking a closer relationship with the EU, arguing such a move could help control immigration and boost growth.
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."
At the Labour conference in Liverpool last month, Mr Burnham said he would set out "options" for Britain's long-term relationship with Europe around the time of a UK-EU summit planned for the end of the year.
Burnham took over as Britain's leader in July and said last month that Britain could consider rejoining the European Union in the future.
It comes after he told The Rest Is Politics podcast, in an interview released on Sunday, that there was evidence Russia interfered in the 2016 Brexit vote, as he warned voters in France and Germany about Moscow-backed efforts to influence their politics.
Burnham is the first British prime minister to openly make such a remark, did not spell out how he believed that Russia interfered in the referendum and gave no details on what evidence the statement was based.
"I asked the question coming to Downing Street, was there interference in the 2016 referendum from Russia? And I've had the clear answer that yes, there was," he said.
The Russian embassy in London did not immediately respond to a request for comment.
Russia has previously denied meddling in the Brexit vote, casting Britain as gripped by anti-Russian hysteria.
Burnham's comments mark a departure from those of former Prime Minister Boris Johnson, who campaigned to leave the EU, and said in 2020 there was no evidence of Russian interference in the referendum.
The UK was plunged into its biggest political crisis since World War II after Britons voted 52:48 per cent to leave the EU.
Burnham said voters in European democracies needed to be better informed about efforts to disrupt their politics.
"These things are important now to put clearly on the record, and I believe, help voters in Germany and France understand that some of this politics is instigated via our enemies, namely Russia," he said.
Speaking to Times Radio on Monday, Foreign Office minister Stephen Doughty declined to provide details of Russia's actions against the UK in 2016.
"But what is clear is that Russia is attempting to interfere in democratic processes, not just in the UK but across Europe," he said.
Britain's government failed to establish whether Russia meddled in the 2016 referendum, according to a 2020 report by parliament's Intelligence and Security Committee.
The report also found that Russia had sought to influence Scotland's 2014 independence referendum, in which voters rejected independence.
Burnham last month said he could campaign to take Britain back into the EU at the next general election, reopening the country's divisive debate on its relationship with the continent.
With Reuters
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