Schools in England could remain closed beyond the middle of January, the education secretary, Gavin Williamson, has admitted, as the country’s largest education authority urged its schools to remain closed and headteachers called for next summer’s exams to be reassessed.

Only three days after declaring he was “absolutely confident” all schools would reopen, Williamson said secondary schools within the 60 “contingency areas” named by the government may remain closed to most pupils after 18 January.

“Those in exam years will be educated remotely during the first week of term, and face-to-face beginning on January 11, with other secondary schools and college students returning full time on January 18, in areas where we have not had to apply the contingency framework,” Williamson wrote in the Mail on Sunday.

The education secretary’s comments are a significant shift from his statements on 31 December, when, asked to rule out further closures, he told the BBC: “We are absolutely confident that all schools are returning.”

Williamson’s remarks came as Birmingham city council – the UK’s largest local authority – asked him to allow the city’s primary schools to close for the next two weeks, and said it would back any schools that defied the government and remained closed.

“The new strain of the virus and the rising case levels in the city mean we are deeply concerned about the return of children to primary schools, special schools and alternative provision next week,” Ian Ward, the leader of Birmingham council, and Jayne Francis, the council member for education, told Williamson.

Boris Johnson also cast doubt on his government’s timetable for schools reopening, announced just five days ago. He told the BBC’s The Andrew Marr Show: “We have got to keep things under constant review.”

Speaking of “tougher measures” being considered to curb the spread of Covid-19, the prime minister said: “Clearly school closures, which we had to do in March, is one of those things.”

On Saturday, all four of England’s main teaching unions, including the Association of School and College Leaders and the National Association of Head Teachers, called for schools to remain closed to all but eligible pupils.

The move was backed by Unison and GMB, representing the majority of other school staff, and by Brighton and Hove council, which called for schools in its area to close for the next two weeks. Unison told its members: “It is unsafe for you to attend the workplace at present if your school is fully opened to all pupils.”

Johnson also suggested for the first time that national exams such as A-levels and GCSEs may not take place in England this summer, telling Marr: “We’ve got to be realistic about the pace of which this new variant has spread … and we’ve got to be humble in the face of this virus.”

The Worth Less? campaigning group of headteachers said the government needed to urgently rethink its approach to exams. It said: “Public safety should not to be risked or driven by an inflexible pursuit of GCSE and A-level examinations in their current form.”

From Monday, the first official day of the new term in England, all mainstream secondary schools and colleges will be closed other than to vulnerable children and those with parents who are key workers. In the 60 contingency areas – hurriedly expanded from 50 after complaints from London local authorities – primary schools will be similarly closed until 18 January.

Dr Susan Hopkins, Public Health England’s senior medical adviser, said: “Attending school is important for the mental health and educational benefits for children. School closure can reduce transmission but the public health advice remains that they should be the last to close and the first to reopen.

“Where rates are extremely high, continuing to rise and the NHS is under significant pressure, it may be necessary to move to remote learning as a last resort.”

Andy Burnham, the mayor of greater Manchester, said the government needed to allow “local flexibility” in deciding which schools to open to most pupils.

“Let headteachers come to a balanced judgment based on what’s happening,” Burnham said.

The Liberal Democrats called for all primary schools to move to remote learning until 18 January, except for vulnerable children and the children of key workers, and for a review of the government plans for Covid testing in schools.