Police Scotland are carrying out further inquiries into an alleged breach of coronavirus rules by MP Margaret Ferrier.
It comes after the force submitted an “initial assessment” to the country’s prosecution service.
Ms Ferrier had the SNP whip removed after it emerged she made a trip from Glasgow to the House of Commons while waiting for results of a Covid-19 test and a return journey after being informed she had the virus.
In a statement, Police Scotland said: “We have submitted an initial assessment of the circumstances to (the) 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 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.
An SNP spokesman said: “Police Scotland and the parliamentary standards commissioner are still considering matters.
“It is important that due process is followed, and we will not be offering further comment.
“Ms Ferrier remains suspended from the SNP.”