Ms Ferrier said she took a test on Saturday afternoon after experiencing “mild symptoms”, meaning she should have self-isolated, before travelling by train to London on Monday.
The MP then spoke in the coronavirus debate in the House of Commons on Monday and, by her account, tested positive for Covid-19 that evening.
“This related to her actions earlier this week, including a train journey on Tuesday, 29 September, between London and Glasgow, following a positive Covid-19 test.
“Following consultation with Police Scotland, officers from the Metropolitan Police, working with British Transport Police, are conducting an investigation into potential offences.”
The Met added that the Parliamentary Commissioner for Standards has also been informed.
The MP for Rutherglen and Hamilton West had the whip suspended on Thursday after she issued a statement apologising for her actions.
“Despite feeling well, I should have self-isolated while waiting for my test result, and I deeply regret my actions,” she said.
“I take full responsibility and I would urge everyone not to make the same mistakes that I have, and do all they can to help limit the spread of Covid-19.”
Nicola Sturgeon called on Ms Ferrier to step down as an MP due to her “dangerous and indefensible” actions.
Scotland’s First Minister tweeted: “I’ve spoken to Margaret Ferrier and made clear my view that she should step down as an MP.
“I did so with a heavy heart – she is a friend and colleague – but her actions were dangerous & indefensible. I have no power to force an MP to resign but I hope she will do the right thing.”
The SNP’s Westminster leader, Ian Blackford, also implied his colleague Ms Ferrier should resign. He said Ms Ferrier should “reflect on her situation”.
The Commons said that she did not inform her party whip until Wednesday afternoon and that one person was identified as a close contact and told to self-isolate.
An SNP spokeswoman insisted the party did not know until Thursday that Ms Ferrier had taken a test prior to travelling to London.
Scottish Conservative leader Douglas Ross, who resigned from Boris Johnson’s Government over Dominic Cummings’ lockdown movements, said that “the public deserve clear answers”.