The warning comes as coronavirus rates have started increasing in some parts of the region with fears of a third wave.
Dr Lisa McNally, Sandwell Council’s director of public health, said: “On very rare occasions we have had a child display classic symptoms but then the parents have refused to have the child tested.
“When this happens we have usually been able to persuade them a test is in everyone’s interests.
“However, if that doesn’t happen then we can’t simply assume the child does not have Covid-19 and that there is no chance of onward infection.
“Therefore, after a risk assessment with Public Health England, we may need to put in place the appropriate outbreak prevention actions.
“Not isolating could risk a large outbreak as seen elsewhere in the Midlands.
“The consequences of this could be potentially fatal and we cannot afford to take a risk.”
More than nine in 10 pupils attended schools in England last week – but a growing number of children are self-isolating at home due to possible contact with Covid-19 cases.
Dr Richard Harling, director for health and care at Staffordshire County Council, added: “Anyone showing symptoms needs to get tested. If this proves to be positive, the school will isolate class or year group bubbles.”
Health chiefs said some schools have faced issues with parents refusing for children to be tested causing disruption for class bubbles and even whole year groups.
It comes as a survey conducted by the Association of School and College Leaders (ASCL), which works with more than 21,000 school and college leaders nationally, found more than half of secondary headteachers had faced difficulties getting parental approval.
A total of 52 per cent of respondents indicated that gaining parental consent has been a difficulty for schools since pupils returned to face-to-face teaching. Geoff Barton, general secretary of ASCL said that this is due to ‘misinformation.’
Now a renewed call is being made for everyone to back the testing effort in the Black Country and Staffordshire.
Catherine Driscoll, director of children’s services at Dudley Council, said: “We urge all secondary school pupils to undertake lateral flow tests and report their results as this helps track the virus and prevent its spread.
“In the case of a pupil receiving a positive lateral flow result or showing symptoms of coronavirus, they would be asked to self-isolate along with everyone in their bubble and household as per government guidelines.”
Levels of consent to Covid-19 testing varies between each school. Chris King, chief executive of the Severn Academies Education Trust, which includes Baxter College in Kidderminster and The Stourport High School and Sixth Form College, said more than 90 per cent of families have signed up to lateral flow tests there.
Mr King described these tests as a ‘key control measure’ to minimise the spread of the virus.
A Wolverhampton Council spokesman said: “If a pupil, or parent, refused to consent when they have symptoms, they and their household would be required to isolate for 10 days.”
Staff in school settings are also being advised to be tested at least twice per week.