The proposal would mean another two- to three-week lockdown, including the October half term, with the closure of hospitality businesses.
The measure was suggested by scientists on the Government's SAGE committee in September, but the Prime Minister instead opted for the three tier Covid restriction system brought in yesterday.
With Shropshire in the lowest tier it means the restrictions are as they were previously.
Councillor Shaun Davies, Labour leader of Telford & Wrekin Council, said he was in favour of the 'circuit breaker' move, as a means to get on top of rising numbers of coronavirus cases.
It came after his party leader, Sir Keir Starmer, announced his support for the 'circuit breaker' plan.
Councillor Davies said: "I think the time has come for a circuit breaker, that is what the scientific advisors were advising three to four weeks ago. We have seen the number of cases rising significantly in Shropshire and Telford & Wrekin in recent weeks."
The council leader said the break would need to be accompanied by more economic support for businesses affected.
He said: "The circuit breaker is not the solution in itself but it gives the government time to get the test, trace and isolate system sorted and reduce the number of cases so they can effectively track and trace again.
"At the moment in Telford & Wrekin and around the country we are seeing cases rising so the restrictions are not working.
"But it is critical that any circuit breaker has to be accompanied by the economic package to support business and people."
He added: "The question is if not a circuit breaker then what? Hospital admissions are going up, sadly deaths are going up, and at the moment there seems to be no real plan to get the numbers down. At the moment the virus is out of control."
The suggestion comes as neighbouring Shropshire Council said that 222 positive cases had been confirmed last week – compared to 29 at the same time in the previous month.
Conservative MP for Shrewsbury & Atcham, Daniel Kawczynski said that the government's new approach should be given time to work.
He said: "I am very conscious of the fact that Shropshire is at the moment in the low category and I feel that Shropshire has done the right thing, Salopians have followed the guidelines and I would have hesitation about another national lockdown, which would adversely impact on Shropshire.
"I think we have to allow the new system the Prime Minister has brought in – where areas are assessed on the number of cases they have – to take effect."
His view was echoed by Montgomeryshire MP Craig Williams, who has seen his own constituents warned they could face a national lockdown from the Welsh Government.
He said: "I don't think we should lock down Powys in its entirety, it is way too vast. If Brecon has an outbreak that is 80 to 90 miles from Llanfyllin, it is madness to group them together.
"I think the same logic applies with a national lockdown. There is no reason Powys and Ceredigion should be locked down because there is an outbreak in South Wales."
He added: "I want to treat the virus in this fire-fighting fashion, which is if you have any spikes in a region, you target it, see if you keep the R number down, and review it in the coming days and weeks to see if it works."
Mr Williams said the public need to follow the rules and do their bit to keep cases down.
He added: "The onus is on the Great British public to apply common sense. There is a real partnership here and if it does not work we will be looking at a circuit breaker which will be bad for the economy and people's mental health."