Jio, BSNL and Vodafone are among those to have adopted the move after a request from the government.
The message, in English, begins with a person coughing and is heard instead of a ringing tone by the caller.
However, several websites have published a way to silence it, after complaints it was becoming annoying.
It contains information about the virus and its symptoms, and is heard only by the caller - it is not a default handset ringtone.
"It's been just about a week since we started hearing this ringtone on many calls," said Kinjal Pandya-Wagh, a senior journalist in the BBC's Delhi Bureau.
"It has helped people understand a bit more about the symptoms and what to do if they have them."
However, she also said it was creating "a bit of panic" among some of those already very anxious about the virus.
Its introduction from 9 March had not been accompanied by much media coverage but had "raised a few eyebrows", Pandya-Wagh said.
The following day, various websites pointed out that by pressing either the number one or the hashtag key, the ringing sound would return instead.