It was expected that the move could be announced before the weekend.
But talks have paused and are expected to resume on Monday.
Speaking on behalf of local leaders, Cllr Mary Lanigan, Head of Redcar and Cleveland Council, said: “We have ongoing discussions with the government about its intention to bring Tees Valley into level three restrictions. .
“Nothing has been agreed and we anticipate further discussions on Monday.
“We will continue to put the health and well-being of our residents first, supporting businesses and saving jobs while pushing for the best possible results for the Tees Valley.”
Teesside has been among the country’s coronavirus hotspots for a month, but things are getting worse.
Level 2 restrictions prohibiting households from mixing outdoors were introduced for Middlesbrough and Hartlepool earlier this month before being extended to Stockton, Redcar and Cleveland and Darlington two weeks later.
Local leaders had demanded that Teesside be given time to see if the Level 2 rules would bring business down.
But the situation has changed rapidly since the start of this week amid growing concern over the growing number of hospitalizations.
Infection rates in Stockton and Middlesbrough are among the highest in the world at Level 2, above 400 per 100,000 in Stockton alone.
James Cook University Hospital has been forced to open a second intensive care unit to care for patients severely affected by the virus.
The government, backed by seasoned scientists, warned local leaders on Wednesday evening that urgent action and a Level 3 move were needed.
This was followed by a briefing with Government Minister Robert Jenrick and Deputy Chief Medical Officer Jonathan Van-Tam on Thursday, where the issues Teesside faces were laid bare.
Mr Van-Tam gave a blunt presentation to the panel – telling them about an increase in rates among older age groups on Teesside and the rate of “hot” positive cases.
He also reportedly said there had been a “very large increase” in cases in Stockton – also highlighting “very large increases” in patients in hospital beds across the region.