They are believed to be rabid fans of actor Vijay, whose film the journalist had criticized.
On Friday, Ms Rajendran tweeted after watching Shah Rukh Khan's latest release "When Harry Met Sejal" that it was "worse" than Vijay's Sura released seven years ago, because then she had stayed at least till the interval.
She was bombarded by vulgar tweets and threats. There have been around 45,000 tweets over the past three days and a derogatory hashtag trended on Twitter.
The alleged trolls have also dug out her tweets in the past that they say targeted Vijay. Naveen Nandan, a Vijay fan said "Dhanya has every right to criticise the film both as a journalist and as a public person. But she can't be too sarcastic. In our city sarcasm is not welcome."
Investigators say based on her complaint, they are trying to identify four people who tweet using aliases. A senior officer said: "We are adopting conventional methods to trace these people. Some have also deleted their accounts".
In a statement, actor Vijay who is known to have political aspirations says, "I respect women. I appeal to not hurt sentiments of women on social media. There's freedom of expression to criticise anyone's film".
The police have booked a case under the IT Act and the Indecent Representation of Women Act among others.