The Scottish National Party leader is expected to make the announcement at a hastily-arranged news conference in Edinburgh.
It is not thought that her departure will be immediate, allowing time for a successor to be elected.
A source close to Ms Sturgeon - the longest-serving first minister - told the BBC that she had "had enough".
Former SNP MP Stephen Gethins said he was "surprised... and disappointed" by the news.
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"Nicola Sturgeon is someone who has led with distinction," he said.
Mr Gethins said it was a "hard, stressful job" and that there would be "big shoes to fill".
SNP MP Stewart McDonald described the first minister as "the finest public servant of the devolution age".
Alison Thewliss, the SNP MP for Glasgow Central, said she was "absolutely gutted" and described Ms Sturgeon as "an incredible leader".
Ms Sturgeon has been a member of the Scottish Parliament since 1999, and became the deputy leader of the SNP in 2004.
She has been first minister since November 2014, when she took over from Alex Salmond after the defeat in the independence referendum.
Ms Sturgeon has led the SNP to a series of election victories at UK, Scottish and local level.
Last year the UK Supreme Court ruled that Holyrood did not have the power to stage another independence referendum - a move which has been blocked by the UK government
Ms Sturgeon wants the SNP to fight the next general election as a de facto referendum, but there has been some opposition to the plan within the SNP.
In addition, recent months have seen controversies over gender reforms, which have been blocked by the UK government.
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