Labour has accused Boris Johnson of deliberately trying to delay the local elections beyond May for his own advantage.
Downing Street is considering plans to delay the elections until June, arguing it would be too dangerous to proceed.
The elections had been set for May 6 and will take place alongside the mayoral elections – which were originally scheduled for a year before.
It would be the first electoral test of the parties and their leaders since the 2019 general election.
But a Number 10 spokesperson said the date “continues to be kept under review”.
Now Labour has accused the Tories of deliberately trying to delay the elections to secure a vaccine “bump” in the polls.
A senior Whitehall source told the Sunday Times: “Local government leaders have had a suggestion from government that the elections are moved to June or July, but many think that’s too soon to ensure polling booths are safe so have mooted the idea of September.
“Holding the elections with postal votes only has been ruled out.”
Shadow local government secretary Andrew Gwynne said: “Countries across the world have held elections. But sadly the Conservatives are now using coronavirus as a justification for fixing the dates of ours to benefit them.
“This isn’t about the pandemic, it is about Boris Johnson trying to get a vaccine bump at the next set of elections”.
A Cabinet Office spokeswoman said: “Legislation provides that the elections will go ahead in May 2021.
“We continue to work closely with the electoral community and public health bodies to resolve challenges and ensure everyone will be able to cast their vote safely and securely and in a way of their choosing.
“Measures are planned to support absent voting at short notice. Guidance will be published in good time ahead of the polls and this matter will be kept under review.”