Natalie McGarry, who sat as an SNP MP for Glasgow East between 2015 and 2017, made the allegations on her private Twitter profile in May in posts which have been seen by The Scotsman.
In the tweets, the former MP claims that Ms Sturgeon “had to stop the FM in his tracks” after Ms McGarry informed a member of the-then deputy first minister’s staff of the late 2013 incident.
Ms McGarry claims she informed this staff member that the incident took place in early 2014 – three years before Ms Sturgeon said she first became aware of the incident.
A spokesperson for the First Minister denied the claims and said the first the SNP leader knew of the allegations against Mr Salmond was in November 2017.
The Scottish Conservatives are demanding answers into the allegation, saying the public “deserve to know the full truth” on the matter.
The allegations come just months after Nicola Sturgeon was cleared of breaching the ministerial code due to her handling of sexual harassment complaints against Mr Salmond.
The First Minister came under significant pressure to resign over the scandal and a parliamentary inquiry into the botched handling of the complaints concluded she had misled Holyrood on the issue of whether she offered to intervene in the process.
Mr Salmond successfully challenged the harassment complaints process through a judicial review, with the handling of the complaints deemed to be “tainted with apparent bias” by the Court of Session.
The action cost the Scottish Government at least £500,000 in costs and the former first minister has indicated he is considering legal action against permanent secretary Leslie Evans for her role in the scandal.
Mr Salmond was also cleared of sexual offence charges in a trial in March last year.
Ahead of the judicial review, it emerged the former first minister apologised to a female civil servant for an alleged incident in Bute House in late 2013.
However, a former SNP MP has claimed she informed a key member of Ms Sturgeon’s staff about one allegation against the former first minister in early 2014.
Ms McGarry took to Twitter, where she holds a private account, to make the allegation, stating the person she told “should have” informed the then-deputy first minister.
In tweets disclosed to The Scotsman, Ms McGarry states that she informed a senior member of Ms Sturgeon’s office of alleged inappropriate behaviour by the former first minister.
She tweeted: “Here is a story. An absolute story. And I kept it quiet until after this election. I knew about one of the worst of the civil service allegations against Alex Salmond at the start of 2014. Happened late 2013.
"I cannot guarantee [they] told Nicola Sturgeon, but I actively told this person she had to. It had to be dealt with.
"If I, as a non-press or civil service person knew about [it], it was out there. And the DFam [sic] had to stop the FM in his tracks. Heard nothing back about or asked.”
In another tweet posted on May 9, the former MP repeated the allegation.
She said: “I voted SNP. I actually hate them though on so many levels. I told NS office manager about one of the key civil service Alex Salmond cases at the beginning of 2014.
"Asked that person to tell NS. I can’t swear that that person told NS. But that person should have.”
The former MP was suspended from the SNP when allegations she embezzled £25,000 from an SNP regional association and the Women for Independence group.
Ms McGarry is awaiting trial in Glasgow for embezzlement and denies the allegations.
However, the former MP’s claim calls into question the stance of the First Minister that the first she knew of any of the alleged incidents was in November 2017 when the Scottish Government investigation into the allegations began.
This claim was also questioned by the Salmond inquiry in their report published in March.
In their highly critical report, the harassment complaints committee stated: “The committee finds it hard to believe that the First Minister had no knowledge of any concerns about inappropriate behaviour on the part of Mr Salmond prior to November 2017.
"If she did have such knowledge, then she should have acted upon it. If she did have such knowledge, then she has misled the committee."
Reacting to the allegations, Scottish Conservative chief whip Stephen Kerr demanded answers from the Scottish Government.
He said: “If these allegations turn out to be true then there will be serious further questions for the First Minister to answer.
“The public deserve to know the complete truth about any reports the First Minister had received in relation to Alex Salmond’s alleged behaviour and when she was made aware of these.”
A spokesperson for the First Minister denied the claims from Ms McGarry.
He said: “As the First Minister has previously made very clear, she was not aware of such concerns prior to November 2017.”