Matt Hancock said that up to three-quarters of new coronavirus cases in he uK were the Indian variant.
A total of 6,959 cases of the variant have now been confirmed in the UK, Public Health England said.
The figures are up to May 26, and represent a rise of 3,535 on the previous week.
In England 6,180 cases have now been confirmed, along with 702 in Scotland, 58 in Wales and 19 in Northern Ireland.
Speaking at a Downing Street press conference on Thursday, Hancock said the increase in cases of the variant remained focused in “hotspots” where surge testing and vaccinations were taking place.
The health secretary said “more than half and potentially as many as three-quarters of all new cases” are of the variant.
“As we set out our road map we always expected cases to rise, we must remain vigilant,” he sad.
“The aim, of course, is to break the link to hospitalisations and deaths so that cases alone no longer require stringent restrictions on people’s lives.”
Hancock said the vaccination programme was having the effect of “severing” that link.
He said “in total” 13,200 deaths and 39,700 hospitalisations had so far been prevented by the vaccine.
The local areas most affected by the Indian variant of coronavirus continue to be Bolton, Bedford and Blackburn with Darwen.
Seven further areas in England have more than 100 confirmed cases of the variant: Leicester, Sefton, Nottingham, Wigan, Central Bedfordshire, Manchester and Hillingdon.
In England on May 24, there were 98 hospital admissions for Covid, slightly above the seven-day average (88) but down 98% from the second-wave peak.