Local and regional restrictions to tackle the Indian variant cannot be ruled out, ministers say.
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It said “economic and social” restrictions could be re-imposed in parts of England if necessary.
Measures could also include bringing forward second vaccines for some people in the worst affected areas.
Public Health England has recorded 1,313 UK cases of the Indian variant.
The figures released on Thursday are more than double the 520 cases recorded by PHE up to 5 May.
The Department of Health and Social Care said there was “no firm evidence yet to show this variant has any greater impact on severity of disease or evades the vaccine”.
But it also said ministers “cannot rule out re-imposing economic and social restrictions at a local or regional level if evidence suggests they are necessary to contain or suppress a variant which escapes the vaccine”.
Health Secretary Matt Hancock said the government was supporting areas where cases were rising and urged people to “continue to be vigilant”.
“We are monitoring the situation very carefully and will not hesitate to take further action if necessary,” he said.
Earlier, Prime Minister Boris Johnson did not rule out reintroducing regional restrictions, saying: “It is a variant of concern, we are anxious about it.”
But he added there was no evidence to suggest England’s planned lockdown easing could not go ahead.
Surge testing is already taking place in 15 areas across England. This is when increased testing and enhanced contact tracing is carried out in very specific locations to prevent the spread of outbreaks.