It is understood that the number of cases is low.
Health bosses are urging people to continue following the guidance to wear face masks, wash hands regularly and maintain social distancing.
Scientists are keeping a close eye on the spread of the Indian variant across the UK, but there are currently no signs that infection is leading to rising hospital admissions, experts have said.
Fiona Phillips assistant director of public health at City of York Council, said people testing positive for variants of concern are traced quickly and have often recently returned from travelling abroad.
But she warned that not all tests pick up variants so it is crucial that people still follow coronavirus safety guidance.
She said: “Not all tests are processed in labs that can do the full sequencing.
“There’s probably about 30 to 40 per cent of tests in York are processed in labs that can do that sequencing.
“It takes about a week for the full sequencing to be done and once that sequencing is done, if it is identified that it is either a variant of concern or a variant under investigation then that is automatically sent to our colleagues in Public Health England. They will do the contact tracing and follow up with the individual.
“For those cases where we know it’s one of those variants, they are followed up reasonably quickly.
“But there is a risk that people could have one of those variants that hasn’t been done in the lab that had done the full sequencing, so it is something that we still need to be cautious around.
“What we see is that most of those variants are where people have come back from foreign travel.
“As foreign travel opens up more, we do really need people to follow that advice when they return from foreign countries in terms of the quarantining that they need to do. That’s going to be really key in the next step of the roadmap.”