The singer said she contacted officers after she was called at home in Blackburn, 10 times over the weekend.
The man allegedly swore at her and told the Britain's Got Talent star: "I know where you live."
Police have taken a statement from Boyle and said inquiries are ongoing. The 54-year-old has spoken of her concern about the calls.
She told the Scottish Sun: "I'm really worried about getting these threatening calls - I haven't slept much since they started. I have no idea who the person is or why they have chosen to call me."
She added: "The police were very reassuring and saying all the right things.
"But deep down, I am really worried."
A fantastic performance of I Dreamed A Dream on Britain's Got Talent in 2009 helped turned Boyle into an international star, with her first album selling millions of copies.
She has since sung for the Pope, the Queen and at the opening ceremony of the Glasgow 2014 Commonwealth Games.
Last year she revealed she had been diagnosed with Asperger's syndrome which sometimes leaves her feeling "a sense of panic".
A Police Scotland spokesman said: "We received a report of nuisance phone calls received at an address in Blackburn on Saturday, 20 June. Inquiries are ongoing."
Police are investigating a series of nuisance phone calls made to Susan Boyle in West Lothian.