Local politicians have said they expect the new measures could go into effect as early as Wednesday.
Warrington Council in Cheshire has already said it will join Level 3 on Thursday.
Downing Street said the talks were still “ongoing”.
The measures come as some scientists have warned that existing measures will not be enough to curb the rapid growth of infections in large parts of the country.
National statistics bureau chief Professor Sir Ian Diamond said it was too early to say the spread of the disease was slowing, despite some positive signs in the latest data.
The government said 151 more people had died within 28 days of testing positive for Covid-19 on Sunday, bringing the UK total to 44,896, and that there had been 19,790 more laboratory-confirmed cases of the coronavirus, which brings the total number in the UK to 873,800.
Over the weekend, South Yorkshire became the last region to come under Level 3 controls after the region of Liverpool, Greater Manchester and Lancashire.
Meanwhile, the Welsh government has announced that it will review a controversial ban on supermarkets from selling non-essential items in a two-week ‘firewall’ lockdown that began on Friday.
Much of the negotiations between the councils in England and Whitehall centered on financial support programs for the affected areas as well as detailed local controls.
Entering level 3 means pubs and bars must close unless they are serving meals while households are prohibited from mixing indoors or in private gardens and beer gardens.
Local leaders have repeatedly complained that the liquidity offered by ministers is not enough to compensate businesses forced to close their doors and the resulting loss of jobs.
The government insists, however, that the latest leave program announced by Chancellor Rishi Sunak and the support available is among the most generous in the world.
At the same time, the government is increasingly criticized that the NHS Test and Trace service, which was supposed to be the key to controlling the disease, is failing.
Senior Conservative MP Sir Bernard Jenkin appealed this weekend for the organization’s leader – Conservative Baroness Harding – to be sacked and replaced by a military commander.
In a scathing attack, he said there was a “leadership vacuum” at the top that “was destroying public cooperation and compliance”.
He was backed by the Labor Party who said Lady Harding’s position had become ‘untenable’ after the latest weekly figures showed less than 60% of contacts of people who tested positive for Covid-19 had been traced and invited to isolate themselves.
Ministers acknowledged that the service needs to improve, but insist Lady Harding has done “a very good job”.
However, it emerged that officials of the Covid-19 task force were considering the possibility of relaxing the rules for those ordered to self-isolate after coming into contact with someone with the disease due to low levels of compliance. .
Ministers confirmed they are seeking to reduce the time people have to quarantine in their homes from 14 days to 10 days and a week.
Sources said the move reflected concerns that people would not respond to when contacted by the system due to fears they could face a long period of time locked up at home if they did.