Jamie Driscoll said that the Chancellor's promised support for people who are unable to work because of restrictions is insufficient.
The North of Tyne mayor has said his area is not due to be put under the highest level of local coronavirus rules next week.
Jamie Driscoll says he and a number of North East local authority leaders were told by the government that they won't be in the top tier of a new system, where it is expected pubs and bars would be forced to close.
It had been thought likely Newcastle would be included in the top level of restrictions, because the rate of recent cases in the city is among the top five areas in the country.
Driscoll, along with three other Labour mayors in the north of England also said earlier that the Chancellor's promised financial support for people who are unable to work because of restrictions is "insufficient."
Greater Manchester Mayor Andy Burnham said that accepting Chancellor Rishi Sunak's financial package would be to "surrender" people to hardship in the run up to Christmas.
Speaking at a press conference with political leaders from Liverpool, Sheffield and Tyneside on Saturday, Mr Burnham said the measures risked "severe redundancies" and business closures.
His comments come as Prime Minister Boris Johnson is set to outline a new three-tiered system of restrictions on Monday (12 October) with measures expected to see pubs and restaurants shut across the north of England.
Under the three-tier system, different parts of the country would be placed in different categories, with areas in the highest level expected to face tough restrictions such as hospitality venues closing.
Mr Sunak announced on Friday that workers in businesses which are forced to close under the new restrictions will have two thirds of their wages paid by the Government.
In an open letter published alongside the press conference the leaders added:
"We believe the Government should bring forward a separate vote on the financial package to provide an opportunity to reject the current financial package and requiring the Government to return with an improved package taking account of the important points we have raised.
"We would ask that you use whatever routes might be open to you to bring about a vote in the House."
The letter is signed by Mr Burnham, Liverpool City Region Mayor Steve Rotheram, Sheffield City Region Mayor Dan Jarvis, Mayor of North of Tyne Jamie Driscoll, and Sir Richard Leese, leader of Manchester City Council.