A photo was tweeted on Thursday of the First Minister with her feet on a couch in her Bute House residence writing the Section 30 letter.
MSPs voted by 69 to 59 this week in favour of seeking permission for an independence referendum to take place between autumn 2018 and spring 2019.
Sturgeon said her mandate for another vote was “beyond question’’, and warned it would be “democratically indefensible and utterly unsustainable’’ to attempt to stand in the way.
Scottish Secretary David Mundell has already said the UK Government would decline the request but Sturgeon is pressing ahead with a formal approach for a Section 30 order - the mechanism for the powers to hold a referendum.
The contents of the letter are due to be made public on Friday.
About 62% of Scottish voters backed the UK remaining part of the EU in June 2016 and the SNP manifesto for last year’s Holyrood elections made clear another ballot on independence should take place if there were a “material change in circumstances’’ from the previous ballot in 2014.
The example cited was for Scotland to be removed from the EU against its wishes.
Scottish Conservative, Labour and Liberal Democrat politicians oppose another referendum and the Prime Minister has repeatedly said “now is not the time’’ for another vote, indicating she will reject the SNP’s preferred timetable.
After the vote in the Scottish Parliament, Sturgeon said she would set out her next steps to Holyrood after the Easter recess if she is rebuffed in the Section 30 request.
The vote followed a meeting between Sturgeon and Theresa May in Glasgow on Monday, and came the day before the process for leaving the European Union was formally triggered.