The Section 30 letter is the formal request to the Westminster government for the power to hold a referendum north of the border.
In a vote on Tuesday, the MSPs voted 69 to 59 in favour of launching indyref2.
On Thursday evening, the Scottish Government tweeted out a picture of Ms Sturgeon working on a final draft of the letter at Bute House.
Theresa May has already indicated that the UK government would block a referendum until the Brexit process has been completed.
The Prime Minister met with Nicola Sturgeon on Monday, when she reiterated that ‘now was not the time’.
Mrs May formally triggered Article 50 on Wednesday, beginning the two-year process of the UK’s withdrawl from the European Union.