The Daily Telegraph said she could name the date for a referendum as early as this week, if Theresa May does not agree to her wishes.
Ms Sturgeon has previously hinted that autumn 2018 would be a suitable time to call a referendum.
Her ultimatum is expected to be given on Monday morning with, the paper said, the intention of influencing a Commons vote later in the day.
If the Article 50 Bill triggering the exit from the EU is passed, it could win royal assent as early as Tuesday, allowing Mrs May to begin the two-year Brexit process.
Ms Sturgeon is believed to want the Prime Minister to include a series of demands for Scotland to be given special treatment in negotiations.
Downing Street has refused to give in to her demands to alter plans unless it faces parliamentary rebellion over the issue.
Ms Sturgeon, the Daily Telegraph said, is understood to be holding a press conference today.
Number 10 said it was unaware of any event taking place.