Council leaders in Greater Manchester say they are ready to meet the PM at any time to try to end the stalemate over bringing in the top level of coronavirus measures for the region.
They want the government to consider an alternative approach to the blanket closure of pubs and bars.
On Friday, Boris Johnson warned he would impose the extra restrictions if no agreement was reached.
But one council leader said he did not expect a meeting before Monday.
Meanwhile, there are more calls for a short but more severe national lockdown – known as a circuit-breaker – across England.
Labour and Greater Manchester Mayor Andy Burnham have urged ministers to bring in nationwide restrictions in the hope this will avert the closure of sectors such as hospitality longer term.
And Britain’s largest teachers’ union, the National Education Union, has called for secondary schools in England to shut for two weeks at half term, rather than the traditional one week.
More than half of England – in excess of 28 million people – is now under extra coronavirus restrictions.
Lancashire has agreed to move into tier three – the highest level – meaning pub closures and bans on households mixing indoors, in private gardens and more outdoor venues. Liverpool City Region was the first of England’s areas to come under the tightest restrictions.
Elsewhere, people in London, Essex, York, Elmbridge, Barrow-in-Furness, North East Derbyshire, Erewash and Chesterfield are in tier two, meaning they can no longer mix inside with those from other households, including in pubs and restaurants.
Mr Johnson warned on Friday that the situation in Greater Manchester, which is currently in tier two, was worsening and that he may intervene if new measures cannot be agreed with the region’s leaders.
But Mr Burnham said more financial aid was needed for workers affected.
Instead of closures, the region’s leaders want measures such as shielding to be considered, and stronger penalties on venues that flout the rules.
“We firmly believe that protecting health is about more than controlling the virus and requires proper support for people whose lives would be severely affected by a tier three lockdown,” Greater Manchester’s deputy mayors and council leaders said in a joint statement.
“We can assure the prime minister that we are ready to meet at any time to try to agree a way forward.”
Rochdale council leader Allen Brett told BBC Newsnight that if Mr Johnson imposed tougher restrictions “then we will have to accept it” but added: “What we want is proper negotiations.
“I stood by all day waiting for a meeting which never took place – I understand now it will take place on Monday – and we need reassurances that this tier three will be time limited and will be effective and I don’t think his scientists think it will.”
Meanwhile, county councils in England are calling for the government to give local authorities more control over the test-and-trace system.
The County Councils Network, which represents local authorities in mainly rural parts of the country, hopes to avoid the prospect of greater restrictions in these areas.
It comes as the PM said on Friday that the UK was developing the capacity to manufacture millions of tests that could deliver results in just 15 minutes.
The new tests are “faster, simpler and cheaper”, Mr Johnson said, adding that work is being carried out to ensure they can be manufactured and distributed in the UK.
The government has set a target of 500,000 tests a day by the end of the month, but the Times reports that scientists expect the UK could carry out a million tests a day by Christmas.