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.”
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.