In 2010 and 2011 some 13 property transactions took place involving Thomson or M&F Property Solutions, a firm she was reportedly said to be a partner in.
The investigation follows a decision by the Law Society relating to these property transactions that resulted in Thomson’s solicitor Christopher Hales being struck off last year for professional misconduct.
In a statement Thomson said she would "cooperate fully" with the police inquiry if required.
She said: "I have always acted within the law and look forward to being cleared of any wrong doing.
"I have this afternoon decided to withdraw from the party whip whilst an investigation takes place.
"Once the investigation is concluded I look forward to returning to play a full role in party activities."
Thomson stood down last night as party whip. She was also formerly the SNP’s Westminster spokesman on business, innovation and skills.
A Police Scotland spokesman said: "Police Scotland can confirm that as a result of a complaint from the Scottish Solicitors' Disciplinary Tribunal, it has been instructed by the Crown Office to carry out an initial investigation into alleged irregularities relating to property deals in the year 2010-2011."
Lorna Jack, chief executive of the Law Society of Scotland, said: "If the Law Society has concerns about any potential criminal matter arising from a SSDT finding, it will refer the matter to the appropriate authorities.
"In the case of Christopher Hales, we first raised this informally with the Crown Office in December 2014. Our Guarantee Fund sub-committee referred it formally to the Crown Office in July 2015."
News of the investigation followed reports in The Sunday Times that Thomson’s property company had bought houses and flats from distressed and desperate sellers at knock down prices.
