Rajiv Shukla said the touring cricketer was in isolation for the last eight days. No other player had been infected, he added.
The player will not be travelling to Durham, where the team will go in a bio-bubble ahead of the upcoming Test series with England.
The two sides begin a five-match Test series on 4 August.
"Yes, 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," Mr Shukla, vice-president of Board of Control for Cricket in India (BCCI), told PTI.
Earlier reports said the cricketer is "largely asymptomatic" and is in "quarantine at an acquaintance's place".
The news came amid reports that the secretary of 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 Mr Shah had told the players to "avoid" crowded places, and going to Wimbledon and the Euro Championships, which recently concluded in the UK.
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 richest franchise tournament in world cricket, was suspended after an increase in coronavirus cases among players.