Congress Party president Sonia Gandhi says those responsible for text messages that caused panic among migrant workers and students in the south of India must be punished.
Some 30,000 people have fled the cities of Bangalore and Mumbai since Wednesday.
India on Saturday blamed Pakistan as the source of text messages and social media postings that warned of reprisal attacks on the families of migrant workers from Assam state in north-east.
At least 78 people have died and more than 300,000 have been displaced in clashes in Assam, mainly between the Bodo tribe and Muslim settlers.
The messages urged people to return to protect their families from further attacks.
The BBC reports many have done so on special trains from Mumbai, Chennai and Bangalore.
Some 250,000 people from the north-east live and work in Bangalore.
On Friday, the government imposed a ban on bulk text messages to try to stop the panic.