Officers had previously been assessing complaints made to them around an “alleged financial irregularity”, but in an updated statement today confirmed a full investigation has been launched.
The SNP said it will cooperate fully with the investigation.
The allegations surround around £600,000 of donations to the SNP which had been part of a crowdfunder for money to be spend on a future independence referendum.
The SNP’s treasurer, Colin Beattie, announced “amounts equivalent” to the money raised would be spent on campaigning for a second referendum following weeks of pressure on the party to identify the money in its accounts.
The funding controversy led to the former treasurer, Douglas Chapman MP, and high-profile MP Joanna Cherry step down from their positions on the party’s national executive committee earlier this year due to concerns around transparency.
A spokesperson for Police Scotland said a full investigation had now been launched into the allegations after seven complaints in relation to donations to the SNP were made.
Prior to this, officers had said they were continuing to “assess” complaints but had not begun investigating the allegations.
A spokesperson said: "Police Scotland has now received seven complaints in relation to donations that were made to the Scottish National Party. After assessment and consultation with the Crown Office and Procurator Fiscal Service, we will now carry out an investigation.
"Enquiries are continuing and anyone who has any information which may assist with this investigation is asked to contact police."
Reacting to the news, Scottish Labour’s deputy leader Jackie Baillie said it was crucial that the truth is discovered.
She said: “I very much welcome the news that Police Scotland has launched an investigation into the claims of impropriety regarding donations to the SNP.
"For many months the SNP has been embroiled in internal fighting and recriminations over this most serious of matters.
"The truth must be known, and Police Scotland must be allowed to carry out a thorough and detailed investigation."
Commenting, the Scottish Conservative chief whip, Stephen Kerr, said police must be allowed to explore “every aspect” of the complaints.
He said: “The SNP have failed to be fully transparent over this funding for months. Senior figures have resigned from the party’s executive committee over the matter but the SNP leadership have still not addressed these serious claims.
“The public to deserve to know the truth. It is only right that Police Scotland are stepping up their investigations having received numerous complaints.
“The police must be allowed to explore every aspect of these complaints to ensure they get to the bottom of this increasingly murky situation.”
A spokesperson for the SNP said: “We will cooperate fully with any investigation.
"As we have made clear, all sums raised for independence campaigning will be spent on independence campaigning.”