The Harry Potter creator said she felt “very sick” after hearing the news and hoped the novelist would “be OK”.
In response, a user said “don’t worry you are next.”
After sharing screen shots of the threatening tweet, Rowling said: “To all sending supportive messages: thank you police are involved (were already involved on other threats).”
A spokeswoman for Scotland’s police said: “We have received a report of an online threat being made and officers are carrying out enquiries.”
Rushdie, 75, was set to deliver a lecture on artistic freedom on Friday in western New York when a man rushed the stage and stabbed the Indian-born writer, who has lived with a bounty on his head since his 1988 novel “The Satanic Verses” prompted Iran to urge Muslims to kill him.
Following hours of surgery, Rushdie was on a ventilator and unable to speak as of Friday evening. The novelist was likely to lose an eye and had nerve damage in his arm and wounds to his liver.
Hadi Matar, 24, of Fairview, New Jersey is accused of attacking Rushdie. He pleaded not guilty to charges of attempted murder and assault at a court appearance on Saturday.
Rowling has in the past been criticized by trans activists who have accused her of transphobia.