Mr Burnham had claimed he hoped the UK would rejoin “in my lifetime”, but gradually watered down his position while voting to leave in the constituency.
An official said there was “curiosity and hope” around Mr Burnham but acknowledged domestic politics could make his position more difficult should he become prime minister.
Another source expressed hope that his expected coronation would lead to a period of stability in British politics, after there have been six prime ministers since the Brexit referendum a decade ago.
Against this backdrop, the European Commission recommended pausing negotiations with Sir Keir to wait for a new prime minister and see whether Mr Burnham would respond more cordially to their demands.
Meanwhile, Kemi Badenoch criticized Sir Keir's plan for his successor to be in Downing Street by September 1, saying national security must be the top priority.
“Explain where the defense money comes from”
The Conservative Party leader said on
“It's really important that we make sure the country is governed. Keir Starmer had to resign because his defense secretary said he didn't care about national security.
“Whatever happens now, we must ensure that national security comes first and that means Andy Burnham must explain where the money to fund our defense comes from and he must cut welfare to fund defense.”
Since Sir Keir's announcement on Monday, veterans of Brexit negotiations have called for closer cooperation between Britain and Brussels. Michel Barnier, the EU's chief negotiator, said he did not want to comment on Mr Burnham's possible push to rejoin the EU.
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