Both schools are in the Wakefield district.
"A small number of cases" have been linked to Sandal Castle Primary School and Outwood Academy in Hemsworth, according to local councillors.
In an update posted to the Wakefield South News Facebook page, Coun Nadeem Ahmed wrote: "As per the update earlier in the week, we are now seeing sporadic cases of the Indian variant in the Wakefield district.
"Ahead of any wider communications I want to let you know that a small number of cases have been confirmed linked to two schools, one of which is in your ward, Sandal Castle Primary School.
"The other school is Outwood Academy Hemsworth.
"We have worked closely with both schools to quickly identify and isolate those who are close contacts and reduce the risk of further spread.
"We are also proactively contacting all parents at the affected schools to remind them of the symptoms of the virus, how to get tested, to take up their vaccine when offered it and to continue to follow – Hands, Face and Space – to help to reduce the spread."
A letter home to parents from Sandal Castle headteacher Nichola Russell on May 27 said: "We have been informed by Public Health England (PHE) and Wakefield Council that a Covid-19 case has been identified in the school community that is the new variant, which was first detected in India (B.1.617.2).
"Close contacts were identified and isolated at the time that the initial positive result was confirmed, and we thank parents for their cooperation with that.
"There is a delay between confirming a positive result and identifying the variant with which someone is infected, so we only became aware that these cases were related to the B1.617.2 variant very recently.
"Whilst there is no evidence to suggest this variant leads to more serious illness, it is thought to be more transmissible and we want to do all we can to reduce the spread of the virus in the community and the wider district.
"This is why we are writing to all parents to ask that anyone who has been in contact with someone who has tested positive, or has symptoms (a high temperature, a new, continuous cough and a loss or change to your sense of smell or taste) arranges for a PCR test as soon as possible."