Internet posts, phone text messages and fake video clips are blamed for spreading rumours that Muslims would attack students and workers from India's northeastern region living in Bangalore and other southern cities.
Tens of thousands of people fled back to the remote northeast last week, fearing an outbreak of violence.
The government demanded that Twitter and other social network websites remove what it has described as "inflammatory and harmful" material. It has also blocked some online content and banned bulk text messages.
"If Twitter fails to respond to our request, we will take appropriate action," senior home ministry official RK Singh was quoted as saying in The Times of India newspaper.
"We have asked the information technology ministry to serve them a notice."
The paper added that the government had set a deadline of yesterday for the websites to respond.
Twitter representatives were not immediately available to comment, but both Facebook and Google have this week said they were in communication with the government and already had policies banning content that incited violence.
Kapil Sibal, the information technology minister, on Wednesday expressed frustration at the delay and difficulty in getting responses from US social networking groups.
"When we tell these sites to inquire about the identity [of people posting material], then they say we are out of your jurisdiction, our servers are outside and we are not under the obligation to disclose the identity," he told reporters.
"So this means it is a platform where anyone can do anything," he added.
The threats against the migrants were linked to weeks of clashes in the northeastern state of Assam between the Bodo tribal community and Muslims that have claimed at least 80 lives and displaced hundreds of thousands.
Mr Singh, who represents Assam in parliament, has described the unrest as a threat to the country's "unity and integrity".