Parrikar is being treated at the Lilavati Hospital in Mumbai since February 15.
"We called journalist Harish Volvoikar as a part of an investigation. He was questioned related to the false news that is being circulated," Inspector General of Police Jaspal Singh told PTI.
"He was questioned for a few hours to get the basic information. After the questioning he was let off," he said.
Earlier in the day, the journalist was also stopped from attending the ongoing assembly session when Goa Governor Mridula Sinha addressed the legislators.
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Singh said a case had been registered and the investigation was underway.
"We have to know the origin of the message which was being circulated. We are going through the whole chain of messages," he said.
Volvoikar represents a WhatsApp-based news service, which covers local events.
"This is to bring to your kind notice that some people are circulating false news about our honourable chief minister's health and misguiding the people in my name," Desai, whose name was quoted in one of the reports of the news service, said in the complaint.