Scientists are keeping a close eye on the spread of the variant across the UK, with new figures from Public Health England (PHE) on Thursday expected to show a big rise in cases.
Speaking at a primary school in Ferryhill, County Durham, the Prime Minister said: "It is a variant of concern, we are anxious about it.
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"At the moment there is a very wide range of scientific opinion about what could happen.
"We want to make sure we take all the prudential, cautious steps now that we could take, so there are meetings going on today to consider exactly what we need to do.
"There is a range of things we could do, we are ruling nothing out."
The Scientific Advisory Group for Emergencies (Sage) is holding a meeting on Thursday to discuss the spread of the Indian variant, amid fears it could have an impact on the Government s road map out of lockdown.
Bolton has one of the highest rates of the Indian variant in the UK, thought to be mostly concentrated in the under-25s.
In Blackburn, extra vaccine doses have been secured so all people aged over 18 can be offered the jab from next week amid rising cases there.
Asked if local lockdowns were possible, Mr Johnson said: "There are a range of things we could do, we want to make sure we grip it.
"Obviously there's surge testing, there's surge tracing.
"If we have to do other things, then of course the public would want us to rule nothing out.
"We have always been clear we would be led by the data.
"At the moment, I can see nothing that dissuades me from thinking we will be able to go ahead on Monday and indeed on June 21 everywhere, but there may be things we have to do locally and we will not hesitate to do them if that is the advice we get."
Asked if masks and social distancing would be scrapped, Mr Johnson said more announcements would be made before the end of the month.
He added: "I think we have to wait a little bit longer to see how the data is looking but I am cautiously optimistic about that and provided this Indian variant doesn't take off in the way some people fear, I think certainly things could get back much, much closer to normality."
Downing Street also said officials would not "rule anything out" when asked if the Government was considering surge vaccinations to accompany surge testing in areas with spikes of new variants.
The Prime Minister's official spokesman told a Westminster briefing: "We want to consider all options.
"The meeting is happening with Sage today and should they come out with any further updates on this variant originating in India and the epidemiology in the UK then we will consider it."
Earlier, Professor Steven Riley, from Imperial College London, said whether the road map for England continued on its planned trajectory was "a Government decision" but suggested the UK was currently in a good place.