The Glasgow North MP stood aside from his Westminster role following claims he groped two male researchers at an SNP Christmas party in 2016.
Further allegations have also emerged about claims that Mr Grady, 41, “inappropriately” touched an SNP staff member, then aged 19, in a London pub.
It has since been claimed the First Minister was told about Mr Grady’s alleged behaviour by Alex Salmond at the same April 2, 2018 meeting that he first disclosed details of the sexual harassment investigation into himself.
Speaking at the Scottish Government’s coronavirus briefing on Monday, the SNP leader confirmed that a formal complaint had been received, but refused to elaborate on how long she has known about the allegations.
Asked when and where she first learned of the claims against Mr Grady, Ms Sturgeon said: “I would have had an awareness previously of a concern, but not a formal complaint.
“I’m not going to say more than that because due process requires that an investigation is allowed to take its course.”
An SNP spokesman previously confirmed a complaint had been made, but said the party would not comment further while the investigation was carried out.
Following Ms Sturgeon’s disclosure, Scottish Conservative MSP Annie Wells said: “This bombshell confession raise further serious concerns about Nicola Sturgeon’s morals and her judgement.
“She admits knowing of concerns about this senior SNP MP yet did nothing about it. Not only did she keep a lid on it, but she actively campaigned for Patrick Grady and then reappointed him to the post of chief whip.
“Every passing day brings new headlines of SNP sleaze. They’re rotten to the core and voters are beginning to see it.”
Ms Wells added: “This latest revelation fuels the perception that some SNP politicians enjoy the protection of Sturgeon and her husband, Peter Murrell, the party’s chief executive.
"If she thinks she can fob people off with the excuse that there was no formal complaint, she is mistaken. That just does not wash in today’s world.
“A blind eye was turned and we need to know exactly what Sturgeon and Murrell knew and when.”