Kate Green, the shadow education secretary, is urging Gavin Williamson to put various contingency plans in place, including ensuring pupils are assessed on what they have been taught and reserve papers being made available for all subjects so self-isolating pupils do not miss exams.
The government has said the plan is for exams to take place in summer 2021, with most delayed by a few weeks to allow for extra teaching time.
“The government's failure to get a grip on this pandemic has meant thousands of pupils off school, missing out on essential learning," Ms Green said.
She added: "To ensure 2021 exams are truly fair to all pupils the government should adopt Labour's plan for greater optionality, reserve papers and regional adjustments to ensure no child misses out."
Kevin Courtney, joint general secretary of the National Education Union, said: "Labour is right to point to huge deficiencies in what the government has so far said about exams for next summer.
"Thousands of teachers and hundreds of thousands of 16 and 18-year-olds and their parents deserve better than this."
Last week, the schools minister, Nick Gibb, said: “We have been very clear that exams are the fairest and best way of assessing student attainment, but we are also conscious of the fact that a large number of pupils have suffered a different experience from other pupils up and down the country.”
He said the government wanted to make sure exams are “as fair as possible while also being valid qualifications".
“That is the work we have been doing with Ofqual and the exam boards for several weeks, and we have announced a delay of three weeks to holding those exams to try to free up as much teaching time as possible,” the minister told parliament.
Schools in England reopened to all pupils in September for the first time in months due to Covid restrictions - and have also stayed open during the country’s second lockdown.
However, coronavirus has continued to cause disruption to education, with around one in 10 secondary state school pupils not in school on 19 November due to reasons linked to Covid-19, government figures show.
As many as 761,000 students were self-isolating on that date over a potential contact with a case of Covid-19, according to Department for Education (DfE) estimates.
Wales has decided to scrap next year’s GCSE, AS and A-level exams for the second year running amid coronavirus, with the country’s education minister saying the pandemic made it “impossible to guarantee a level playing field” for students facing exams.
Additional reporting by PA