The Mayor of Greater Manchester has been given until midday on Tuesday to reach an agreement on lockdown restrictions in the region.
Andy Burnham had said he still hoped to reach agreement with the Government.
In a joint statement with Manchester City Council leader Sir Richard Leese, they said it was "surprising and disappointing" that an earlier offer of a hardship fund to top up furlough payments and support the self-employed had been "taken off the table" by Health Secretary Matt Hancock.
"This evening we have written to the Prime Minister reiterating our willingness to continue to work towards an agreement but reminding him that Greater Manchester has been in Tier 2-style restrictions for almost three months, and that this has taken a toll on people and businesses here," they said.
"With this in mind, we do not believe it is in any way unreasonable for us to require better protection for our lowest-paid residents."
They also complained that the Government had today sought to raise public concern about the NHS in Greater Manchester with the release of "selective statistics" about the occupancy rates in intensive care.
But Communities Secretary Robert Jenrick has since said in a statement that he has written to local leaders in Greater Manchester giving them until midday on Tuesday to reach an agreement on the introduction of Tier 3 restrictions.
He said it was not too late for local leaders to work together with the Government.
A statement from Andy Burnham, Mayor of Greater Manchester and Sir Richard Leese, Leader of Manchester City Council and Greater Manchester Health Lead said: “We went into today’s meeting with the Government with a positive and unanimous view amongst Greater Manchester Leaders that we should seek a resolution as soon as possible.
“We had been encouraged by earlier discussions at an official level where the idea of a Hardship Fund, to top up furlough payments and support the self-employed, had been tabled by the Government.
"It was both surprising and disappointing when this idea was taken off the table by the Secretary of State.
“We are appreciative of discussions resuming later this afternoon at officials level and very much hope this will lead to a positive outcome."
They said being in Tier-2 style restrictions for almost three months had taken a toll on people and businesses.
The statement added: "With this in mind, we do not believe it is in any way unreasonable for us to require better protection for our lowest-paid residents.
“Finally, we are disappointed that the Government has today sought to raise public concern about the NHS in Greater Manchester with selective statistics.
“Greater Manchester’s ICU occupancy rate is not abnormal for this time of year and is comparable to the occupancy rate in October 2019.
"Also, providing information about individual hospitals does not reflect that our hospitals work as a system to manage demand.
“We are not complacent about the position in our hospitals and are monitoring the situation closely.
"But in the current situation, we believe it is essential that our residents are given clear, accurate information about the state of the NHS in Greater Manchester and that public fears are not raised unnecessarily.”
But Housing Communities and Local Govt Secretary Robert Jenrick said: "The deteriorating public health situation in Greater Manchester means that we need to take action urgently.
"We have held discussions in good faith with local leaders for ten days in order to ensure that the measures put in place were tailored to the local community.
"We have offered an extensive package of support for local people and businesses, proportionate to the approach we have taken in the Liverpool City Region and Lancashire and in addition to the wider national support.
“There are now more COVID-19 patients in Greater Manchester Hospitals than in the whole of the South West and South East combined.
"But, unfortunately, despite recognising the gravity of the situation, local leaders have been so far unwilling to take the action that is required to get this situation under control.
"I have written to local leaders this evening to make clear that if we cannot reach agreement by midday tomorrow then I must advise the Prime Minister that despite our best endeavours we’ve been unable to reach agreement.
"It’s not too late for local leaders to work with us to take action for the sake of the people of Greater Manchester.”