Burnham went public on Monday with advice he said he received when he became prime minister, after asking about foreign interference in the referendum. The victors of the 2016 Leave campaign cried foul, denouncing the claim as an attack on the legitimacy of the result.
Remain supporters, by contrast, praised Burnham for finally admitting what some observers said at the time was a clear case of foreign interference.
The timing is especially sensitive. Burnham believes Brexit has damaged Britain over the past decade and wants to tighten ties with the EU in talks next month in the hope of lifting growth. Leave campaigners suspect the prime minister will one day call a new vote to reverse Brexit, and see his comments as a step in that direction.
The row has given fresh energy to the long-running debate over whether Britain might eventually rejoin the bloc, an idea opposed by those who campaigned to leave.