The MP for Hamilton and Rutherglen was suspended from the SNP party after she admitted to taking a train from Scotland to London when she should have been self-isolating.
Ms Ferrier was awaiting results after testing for Covid-19 while she was travelling between the two countries.
She then returned home by train after being confirmed as having tested positive for the virus.
In a statement, a police spokeswoman said: “We have submitted an initial assessment of the circumstances to Crown Office and Procurator Fiscal Service and are carrying out further inquiries under their direction.”
The Crown Office declined to comment.
The announcement follows the Metropolitan Police confirming last month they would take “no further action” against the MP after an investigation.
They said regulations under which Ms Ferrier could have been punished were not in effect when she travelled.
Despite facing pressure to resign her seat, Ms Ferrier is now an independent MP for Hamilton and Rutherglen and asked a question remotely at PMQs on Wednesday.
In a statement on October 15, Scotland Yard said the matter had been referred to Police Scotland for consideration after the London force ruled the MP’s coronavirus test occurred before September 29, when new regulations came into effect that could have resulted in a £10,000 fine being levied against her.
At that time, Police Scotland said they were aware of the decision by the Metropolitan Police and would “now assess the circumstances and consult with the Crown Office and Procurator Fiscal Service before taking a decision on next steps”.
SNP leader Nicola Sturgeon and Labour leader Sir Keir Starmer are among those who have called on the MP to stand down.
In an interview with The Scottish Sun last month, Ms Ferrier said she felt she was “hung out to dry” by the SNP, which withdrew the whip from her one hour after she released a statement detailing her actions.
She also said she had no intention of standing down.