India wicketkeeper Rishabh Pant and training assistant Dayanand Garani have tested positive for Covid-19.
Pant, 23, is close to completing his quarantine period and can join the squad in Durham if he returns two negative tests.
The BCCI medical team identified Bharat Arun, Wriddhiman Saha and Abhimanyu Easwaran as close contacts of Garani.
India will enter a bio-bubble before the five-match Test series with England starting on 4 August.
The India players have remained in the UK after the World Test Championship Final defeat by New Zealand last month.
They were given a 20-day break before joining back together.
The news came amid reports that the secretary of Board of Control for Cricket in India (BCCI), Jay Shah, had sent an email to the 23-member team in the UK warning it about rising Covid-19 cases in the country.
Reports said Shah had told the players to "avoid" crowded places, and going to tennis Grand Slam Wimbledon and football's European Championship, which recently concluded in the UK.
Pant shared pictures on social media attending the last-16 Euro 2020 match between England and Germany at Wembley on 29 June.
Rajiv Shukla BCCI vice-president earlier told PTI: "One player has tested positive but he has been in isolation for the last eight days. He was not staying in any hotel with the team, so no other player has been affected."
Earlier this month England named an inexperienced 18-man replacement squad for their ODI series with Pakistan after three players and four staff members tested positive for Covid-19.
In May, the Indian Premier League (IPL), the the richest franchise tournament in world cricket, was suspended after an increase in coronavirus cases among players.
'News of huge concern' - analysis
by Stephan Shemilt, BBC Sport cricket writer
The news of Rishabh Pant contracting Covid-19 is a huge concern for the packed schedule of cricket that lies ahead in this home summer.
Just last week we had an entire squad of England players having to isolate. The ODI series against Pakistan was still able to be played, but what if the same happened to the India team? A Test or perhaps even the entire series - so lucrative and important - would be in jeopardy.
These cases, and outbreaks in the domestic game, also add to nervousness around the inaugural season of The Hundred. What would happen if one of the franchises is struck by multiple positive tests?
And, further down the line, what about the Ashes? We know that Australia has some of the strictest Covid protocols in the world, but only on Thursday did ECB chief executive Tom Harrison say that the players are "fed up" of living in bio-secure bubbles.
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