The Conservative-led British Government, however, must agree to a legally binding referendum, and May said "now is not the time".
"All our energies should be focused on our negotiations with the European Union," May said, adding that holding a Scotland referendum while EU exit talks were still under way would "make it more difficult for us to get the right deal for Scotland and the right deal for the UK".
The British Government's Scotland Minister, David Mundell, said May's Administration "will not be entering into discussions or negotiations" about a new referendum on Scottish independence.
But Sturgeon, undeterred, plans to ask the Scottish Parliament next week to start the process of seeking a new referendum.
She said it would be a "democratic outrage" for the British Government to stop the people of Scotland "having a choice over their future".
"It is for the Scottish Parliament - not Downing Street - to determine the timing of a referendum, and the decision of the Scottish Parliament must be respected," she said.
She said "having sunk the ship with Brexit", the British Government was "trying to puncture Scotland's lifeboats as well".
Sturgeon's Scottish National Party does not hold an outright majority in the Scottish legislature, and her plan will face opposition.
Scottish Conservative leader Ruth Davidson, whose party has the second-largest bloc of seats, said the Tories would "reject conclusively the timetable for a referendum set out by the Scottish Government".
Davidson said voting on independence without knowing how Britain was faring outside the EU would mean that "on the most important decision we can make, we would be voting blind".
Scottish voters rejected independence by a margin of 55 per cent to 45 per cent in a 2014 referendum that was billed as a once-in-a-generation vote. But Sturgeon says the UK's decision to leave the EU has changed everything. AP