Cabinet Office minister Michael Gove made the accusations against Mr Burnham as he called for action “to save people’s lives”.
The Greater Manchester Mayor and Conservative politicians in the region oppose Tier 3 measures being imposed unless there is greater financial support for workers and businesses. Talks between Mr Burnham and Westminster are set to resume today.
Mr Gove told Sky’s Sophy Ridge On Sunday: “I want to reach an agreement with the political leadership in Greater Manchester. I want them to put aside for a moment some of the political positioning that they’ve indulged in and I want them to work with us in order to ensure that we save lives and protect the NHS.
“Instead of press conferences and posturing what we need is action to save people’s lives.”
He went on to say that an "absence of action" would see more people get infected and put more pressure on the NHS.
But Mr Burnham insisted he was looking for a "resolution" as he called for MPs in Westminster to intervene and come with an adequate financial package.
He told the BBC’s Andrew Marr Show: “Let’s move towards a resolution. That’s why I’ll be writing to the Labour Party leaders in Westminster to ask them to intervene, for Parliament to intervene here.
“What we need here is a fair financial framework if the Government are going to insist on Tier 3, at the moment they’re doing side deals with individual councils, that isn’t good enough for me.”
He reiterated his call for the 80 per cent furlough scheme to be reintroduced to support workers in firms that are forced to close. “This is everywhere’s concern because everywhere could end up in Tier 3 over the course of this winter,” he said.
“Support us in this, I would ask your viewers to contact their MPs to support us in this because it’s about taking the country fairly through a very challenging winter. If we don’t put in this support Tier 3 will create a punishing lockdown in the poorest communities.”
He also Prime Minister Boris Johnson of having exaggerated the severity of the coronavirus situation in the region.
“It’s a serious situation but I don’t think it was the situation that was described by the Prime Minister on Friday evening. I think it was an exaggeration of the position that we’re in," he said.
“Of course it’s a matter of concern, and we watch the figures very closely indeed, but the figures have been falling in Manchester itself in the last few days, across Greater Manchester up slightly but certainly not doubling every nine days.
“So let’s be careful here. I would certainly say this morning let’s step back a bit from a war of words.”
Despite the continued war of words, a call was scheduled for Sunday between Mr Burnham and the Prime Minister’s chief strategic adviser Sir Edward Lister after confusion over whether talks would continue this weekend.
Asked if he was going to be speaking with Boris Johnson, Mr Burnham told the BBC’s Andrew Marr Show: “I was due to speak to Sir Ed Lister today so that’s the call I was expecting to take later this morning but beyond that I’m ready to speak to ministers to try to resolve this situation.”
He told the BBC’s Andrew Marr Show: “No, we worked effectively I think and co-operatively with them because there’s a widespread recognition that unless we deal effectively with the virus in those areas where the incidence is high, then we won’t be able to improve the health of all.”
Responding to accusations of the Government “blackmailing” local leaders into doing a deal, Mr Gove said: “It is the case that in every part of Lancashire and in Liverpool, agreement was reached.”
Denying the claims, he added: “No, no, it’s because we – A were very clear about the public health problem; and B provided economic support.”