Some 14 council leaders from across the region have written to the Health Secretary to offer help and press the urgency of the situation.
Mr Hancock said today that: "The message to people is do come forward, there are tests available.
"By the way there are tests available in every part of the country.
"Do not believe these stories when they appear in newspapers saying there aren't tests available here, there are tests available in every part of the country."
And Downing Street rejected claims that people are not able to get coronavirus tests, despite photos and testaments showing otherwise.
Asked about testing levels, a Downing Street spokesman said: “We are ramping up capacity, or working to ramp up capacity in terms of tests.
“I am not aware of anything to suggest that tests are not available in some parts of the country.”
Head of NHS Test and Trace Baroness Harding admitted on Thursday that “it’s clear from today that demand is significantly outstripping the capacity we have” to conduct coronavirus tests.
She said: “We have to restrict the number of people who are taking tests in the testing sites so that there’s no risk of those tests going out of date when they are processed in the labs.”
It comes after dozens of drivers turned up at a test site yesterday to find there were no staff to swab them, on the day the Health Secretary announced tougher coronavirus measures for people in the north-east.
People who had booked a test on Thursday at Doxford Park, an out-of-town business park in Sunderland, were told by the media they would not be tested, as there were no officials there to inform them.
Some had been turned away on the approach to the centre by security guards, who told them the computers had crashed and to try again later.
HGV mechanic Brad Cockburn, 28, made a 100-mile round trip from Bedale, North Yorkshire, only to find there were no staff, not even a tent or other infrastructure, at the site on the out-of-town business park.
He said: “There’s no organisation, it’s p***-poor performance as usual.”
Mr Cockburn’s employer booked him the test as he felt as if he had flu-like symptoms and he is now unable to return to work until he receives the all-clear.
“They’re supposed to put these things in place to get people working again,” he added.
“Now they’ve got all these people congregating here and nobody to test them.”
Now, 14 of Yorkshire’s council leaders have written to the Health Secretary offering to help.
In a cross-party letter the leaders of councils in Barnsley, Bradford, Calderdale, Doncaster, East Riding, Hull, Kirklees, Leeds, North East Lincolnshire, North Yorkshire, Rotherham, Sheffield, Wakefield, and York, said: “There are very immediate issues that are particularly acute in the watch list authorities as we try to manage the capacity problems locally.
“Areas on the watch list were previously assured that their access to testing would be protected but this isn’t our experience in Yorkshire and Humber at this critical moment.”
They said: “People with symptoms are being turned away if they do not have an appointment and when they return home to book an appointment the national portal is not sending the confirmation they need to show testing sites that they have booked an appointment.”
They warned public confidence in the testing system was “fragile” but offered to help by locally organising more testing capacity.