Rachel Spencer-Henshal, Director of Public Health at Kirklees Council, said cases of B.1.617.2 were detected in one household and everyone who tested positive and their close contacts have been told to isolate.
Public Health England states the variant is at least as infectious as the strain first detected in Kent which caused the spike in cases that resulted in the third national lockdown, but there is “insufficient evidence” to determine whether it is more deadly or vaccines are less effective against it.
“We’re being extra cautious and doing everything we can to monitor and prevent its spread,” said Ms Spencer-Henshal.
“Some members of this household are pupils at a local school. So, any pupils or staff at the school classed as a close contact - as well as those they share a household with - are being asked to take a PCR test.
“This will help us to identify any variant of concern cases so that we can contain the spread.”
Kirklees has the fourth highest infection rate in England, of 69.1 cases per 100,000 people, after it recorded 304 confirmed cases in seven days.
Surge testing is being carried out in the three areas with the highest rates - Bolton, Bedford and Blackburn with Darwen - due to concerns about the spread of the Indian variant.
Speaking in Parliament this week, Matt Hancock said 2,323 confirmed cases of that variant have been recorded in the UK, including 483 in Bolton and Blackburn with Darwen where it “is the dominant strain”.
Thousands of extra doses of Covid-19 vaccines have been provided in both areas, to try and curb the spread.
Ms Spencer-Henshal admitted the infection rate “looks to be rising” in Kirklees but deaths and hospitalisations remain at a low level and the vaccine take up is “good”, as more than 340,280 doses have been administered.
Kirklees Council has not asked the government to provide additional testing or vaccine doses yet.
Ms Spencer-Henshal said: “We need to think about capacity and supply before taking a decision like that and we need to remember that it’s just as important that people get their second dose to maximise protection.
“These are all the factors we – and government - need to weigh up. In the meantime, anyone over the age of 36 can get a vaccination and I encourage everyone eligible to get one.”
She added: "It’s difficult to know why our infection rate is as high as it is. However, we do know, based on the types of industries we have in Kirklees, that a larger proportion of our population have to leave their house to work which introduces more circumstances where people have to mix with others.
“We’re at a crucial point in the pandemic and we all have a really important role to play in getting Kirklees through the roadmap out of lockdown safely.
"Whilst deaths and hospitalisations remain at a low level, we have to make sure our infection rate does not increase and lead to more people being admitted to hospital or losing their lives to Covid-19.
"The way we can all keep each other safe remains exactly the same. Please all continue to wash your hands well and often, wear a face covering and make sure you’re social distancing when you’re out. Please also remember that it’s so much safer outdoors if you are meeting with people form outside your household for any reason.
"Crucially, I’m urging everyone to please take the vaccine when it’s their turn. It doesn’t matter how young or healthy you are, Covid-19 still presents a risk and the vaccine protects not just you, but those around you too. Every vaccine matters. This is our main and most sustainable route back to normality."