UK Health Secretary Matt Hancock said the "vast majority" of people in the town, some 280 km from London, who are in hospital with the new variant had not taken up the offer of a vaccine.
He said the government was prepared to implement restrictions if necessary.
Bolton town has seen a sharp spike in infections since mid-April, Public Health England said.
In the town, the B.1.617.2 variant now makes up the majority of new coronavirus cases.
Hancock said the plan was to "pile in testing and vaccinations to try to get on top of this".
He said five people who had received a single jab were in hospital in Bolton with the variant first identified in India.
One patient had received both doses of the vaccine- but they are "very frail".
Asked if anyone has died with the B1.617.2 variant after receiving two jabs, Hancock said: "Not that we're aware of".
"We've put in a huge amount of testing. We've reinforced that with army personnel as well, and we're expanding the vaccination programme - especially for those second jabs which are the best protection against ending up in hospital," Hancock told the Sky News.
On the prospect of further action, he said: "At the moment we're taking the approach that worked in south London - which is this massive surge testing - but of course we don't rule out further action.
"Given though Bolton has been in some form of a lockdown for a year, it's not a step we want to take but of course we might have to take it and we will if it's necessary to protect people."
Leader of Bolton Council David Greenhalgh urged people to get the vaccine.
"We've been knocking on doors," he was quoted as saying by the BBC.
"You can take a horse to water but you can't always make it drink but we are doing everything we can through all our partnerships, through our faith leaders and everyone, to get the message out there: get a vaccine, it protects us all.
"If you've got the offer, the offer is going to be there... please take up that vaccine," the Conservative councillor added.
Attention has been focused on just a handful of neighbourhoods to the south of the town centre - Rumworth, Deane and Great Lever.
About half of new cases detected in Bolton over the past week have come from those three areas alone.
There is currently no evidence it causes more severe disease or can evade Covid vaccines in any way - the single biggest concern of scientists.