Communities Secretary Robert Jenrick said he has written to local leaders in the region giving them until midday on Tuesday to reach a deal on the issue.
Greater Manchester Mayor Andy Burnham earlier said that he still wanted agreement on a financial support package before agreeing to go into the restrictions.
But the Government on Monday night issued an ultimatum, giving the region until lunchtime to reach an agreement.
Mr Jenrick said: "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."
Tier 3 measures would see pubs and bars ordered to close as part of a package of strict measures.
Mr Burnham hit out at Downing Street for using "selective statistics" to raise concern about the public health situation in the region.
It followed a warning from a Government spokesman that the entire intensive care capacity in Greater Manchester could be filled with Covid-19 patients by November 12 unless action was taken.
However, the two leaders insisted the region's intensive care occupancy rate was not abnormal for this time of year.
"We are not complacent about the position in our hospitals and are monitoring the situation closely," the said.
"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."
Mr Jenrick insisted the discussions - which have been going on for 10 days - had been conducted in "good faith".
He said said they had offered an extensive package "proportionate" to the approach taken in the Liverpool City Region and Lancashire.
Mr Burnham and Manchester City Council leader Sir Richard Leese said the region should receive "better protection" for low-paid residents if it is to agree to the restrictions.
In a joint statement, 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".
"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."
Senior Conservatives, including the influential chairman of the 1922 committee, Sir Graham Brady, have backed the Greater Manchester mayor in resisting Tier 3 for the region.
Mr Jenrick earlier said there had been "productive" negotiations with Greater Manchester but Monday's talks would be the "final discussions".
"We do now need to draw these conversations to a conclusion and to take decisive action," Mr Jenrick told BBC Radio 4's Today programme.
"We have set out the action that we think is appropriate but we are also willing to continue, and have final discussions, with council leaders and the mayor in Greater Manchester today to see if we can reach an amicable agreement."
Liverpool City Region received a £44 million package as part of its Tier 3 measures, but Greater Manchester wants more extensive support.
This support being demanded would include a furlough scheme paying 80 per cent of wages for affected workers.
Oldham's council leader, Sean Fielding, one of the local authority chiefs in Greater Manchester, said the row was not just about money but whether being placed in the "very high" alert category would actually have an impact on the number of coronavirus cases.
The Labour politician told Today: "This is about doing the right thing to bring the virus back under control and relieve pressure on the NHS.
"But if the Government does want to impose measures on Greater Manchester that are not with the support of their scientific advisers, then we will not allow our residents and businesses to be left financially worse off by the imposition of these untested measures."
Treasury sources said Chancellor Rishi Sunak would not get in the way of a deal, but said he would not cave in to Mr Burnham's demand for a return to the 80 per cent furlough scheme for people whose workplaces are forced to close.
Mr Burnham said whether a deal can be reached is "not about the size of the cheque".
He told Sky News: "It's about protecting low-paid workers, people who are self-employed, supporting businesses and preventing them from collapsing. That's what this is about.
"We've always said we would put people's health first, and we will do that. But health is about more than controlling the virus - people's mental health, I think, is now pretty low given that we've been under restrictions here for three months already."
Mr Burnham will be under renewed pressure to accept a lockdown after the Guardian reported a leaked NHS document revealed Greater Manchester is set to run out of beds to treat those seriously ill with Covid-19 and that some of the region's 12 hospitals are already full.
Mr Jenrick told Sky News: "The number of cases is rising and the pressure on some of the hospitals in Greater Manchester is there for all to see."
Mr Burnham turned to Westminster to help break the deadlock, writing to party leaders to warn that "most places" will end up in Tier 3 at some point before a vaccine is rolled out.
He said "clear national entitlements" similar to during the first lockdown are essential to create the "sense of fairness" to ensure compliance with new restrictions.
Monday night's developments came after the Welsh government confirmed a two-week "firebreak" lockdown in the country.
It will introduce the shutdown in Wales from 6pm on Friday, First Minister Mark Drakeford said.
And business leaders also demanded help from the Treasury for firms in England forced to close in Tier 3, as well as those in Tier 2 which face "dramatically" hampered finances.