The Harry Potter author, 57, shared screenshots of a message from a user who wrote: “Don’t worry you are next” in response to her tweet about Rushdie last month.
The Indian-born British author, whose writing led to death threats from Iran in the 1980s, was stabbed in August ahead of a lecture at the Chautauqua Institution, 65 miles from Buffalo in New York.
Rushdie has faced years of threats over his book The Satanic Verses, which has been banned in Iran since 1988 as many Muslims view it as blasphemous.
Rowling had said she felt “very sick” after hearing the news and hoped the novelist would “be OK”.
The same Twitter account to have messaged Rowling also posted messages praising the man who carried out the attack. The tweet, which came from an account in Pakistan, was later removed.
A Police Scotland spokeswoman said: “Following a report made to police on Saturday August 13 2022 regarding an online threat, enquiries have been carried out into this matter and it has been established that it was made outwith the UK.
“Enquiries are now complete and there is no further police action at this time.”
Additional reporting by Press Association.