Police are investigating the actions of MP Margaret Ferrier, who travelled from Glasgow to London with Covid-19 symptoms then returned home after testing positive.
The Met said it was looking at possible offences related to the Health Protection Regulations.
Ms Ferrier has been widely condemned for risking the health of people in parliament and on public transport.
She has been suspended by the SNP and faces calls to quit as an MP.
The Met said in a statement that an MP had contacted Police Scotland on Thursday to say she may have breached legislation and guidance relating to coronavirus.
It said this had related to actions including a train journey on Tuesday 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,” said the Met.
“The Parliamentary Commissioner for Standards has been informed.”
Commons speaker Sir Lindsay Hoyle had earlier spoken of his disbelief at Ms Ferrier’s “reckless” actions, which he said had put other people’s health at risk.
And Scottish First Minister Nicola Sturgeon, the SNP leader, said she had spoken to Ms Ferrier and urged her to “do the right thing” and stand down as an MP.
She said Ms Ferrier had been guilty of the “worst breach imaginable” and that her “reckless, dangerous and completely indefensible” actions had undermined the public health message.
DUP MP Jim Shannon, who was seated at the same socially-distanced dining table as Ms Ferrier on Monday evening, is self-isolating but received a negative test result on Thursday afternoon.
An Assistant Serjeant at Arms was close to Ms Ferrier when she spoke in the Commons on Monday but has not been advised to self-isolate.
In an interview with BBC Scotland, Prime Minister Boris Johnson would not be drawn on whether he thought Ms Ferrier should stand down as an MP.
He said it was important that “everyone obeys the rules and guidance”.
Ms Ferrier, the MP for Rutherglen and Hamilton West, travelled by train to London on Monday despite being tested for Covid at the weekend after experiencing mild symptoms.
She spoke in the Commons chamber during a coronavirus debate, but was told later that evening that she had tested positive for the virus.
Despite this, Ms Ferrier took a train back to Scotland on Tuesday, with SNP whips in the Commons being told about her positive test on Wednesday.
It became public knowledge on Thursday evening when Ms Ferrier tweeted an apology and said she “deeply regretted” her actions.
She has not yet given any indication of whether or not she intends to continue sitting as an independent MP.
Source:BBC News – UK