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.