Alex Salmond, the party’s leader, said those who threaten their candidates would be “faced down”.
He also threatened to report media outlets who “fan the flames of this disturbing behaviour” to the appropriate regulator.
The news comes after a social media storm erupted following the Alba Women’s conference.
Speaking at the conference, party candidate Margaret Lynch accused two Scottish LGBT groups, Stonewall Scotland and LGBT Youth Scotland, of seeking to “reduce the age of consent to 10 years old”.
The claims were based on the two groups’ affiliation with ILGA World, and that umbrella group’s connection to a 2020 feminist declaration signed by more than 200 organisations worldwide.
ILGA World, Stonewall Scotland, and LGBT Youth Scotland have all categorically denied the claims, with Lynch accused of “repeating deeply harmful myths about LGBT people as paedophiles and sexual predators” and spreading “vicious lies”.
In a statement issued this afternoon, Salmond claimed that, after being widely publicised over the weekend on social media, Lynch’s comments led to a deluge of threats against Alba women.
His party said it had reported some on social media to the police, and that they were “drawing the line”.
Those that Alba say are “among tweets that have been reported to the police” are a Scottish parliamentary candidate from the SNP and another from the Scottish Greens.
Salmond said: “People who threaten our women candidates will be held to account. Men who put at risk women’s jobs and livelihoods will be faced down. Name-calling of Alba women by male candidates will be challenged, and media who fan the flames of this disturbing behaviour will be reported to the appropriate regulator.
“In an election campaign people often do and say foolish things but we shall not have our female candidates bullied or threatened. That stops right now.
“Apart from anything else Margaret Lynch was defending the rights of women in Scottish society while these keyboard warriors were in nappies.
“Let me be clear. As the First Minister who introduced equal marriage in Scotland, I would never tolerate any homophobic behaviour.
“Women are entitled to raise concerns without being targeted in this fashion. If people wish to disagree then by all means go ahead. But our candidates are entitled to speak their minds at safe space events without being traduced or intimidated. Alba will protect their right to do so. ILGA World has issued a statement providing clarification which I welcome.”
He added: “Now that this particular matter has been reported to the police ALBA will make no further comment on this specific controversy.”
ILGA World’s statement said it was “appalled that false stories are circulating attempting to imply that ILGA World is advocating to lower or eliminate the age of consent through an obtuse interpretation of a paragraph in a group statement made in 2020”.
Nancy Kelley, CEO of Stonewall, and Colin Macfarlane, director of Stonewall Scotland and Northern Ireland, said: “We are appalled by the baseless and offensive accusations made by Alba parliamentary candidate Margaret Lynch.
“What she said was not only untrue, but it was also a wilful, bad faith interpretation made for the sole purpose of trying to smear LGBT+ organisations.”
The Scottish Greens, SNP, and Police Scotland have been approached for comment.