Ms. Rowling had reacted to the attack on the author of "Satanic Verses" by saying she was "disgusted" on the social network, adding that she hoped for her recovery.
A user, who presents himself on his profile as a student and political activist based in Karachi in Pakistan, then replied: "Don't worry you're next".
The tweet was later deleted but JK Rowling posted a screenshot, calling out Twitter for possible violations of its rules.
Threatened with death since a "fatwa" from Iran in 1989, a year after the publication of the "Satanic Verses", Salman Rushdie was stabbed a dozen times on Friday, an attack which outrages in the West but which is hailed by extremists in Iran and Pakistan.
"We have received information about a threat made online and our officers are investigating," said a spokeswoman for police in Scotland, where the author lives.
Last year, J. K Rowling said she had received numerous death threats, she said, from transgender rights activists, of whom she has become a pet peeve in recent years.
In 2020, JK Rowling shared an article on Twitter referring to "people who menstruate", commenting ironically: "I'm sure we had to have a word for these people. Someone help me. Feum ?fame?feemm?.
She thus attracted the wrath of some Internet users, who reminded her that transgender men could have their period, but not transgender women.
© 2022 AFP