Ms Ferrier travelled to the Westminster Parliament despite being required to self-isolate with coronavirus symptoms, and then returned to Scotland by train after testing positive.
The SNP has withdrawn the whip from the Rutherglen and Hamilton West MP and First Minister Nicola Sturgeon and other senior figures in the party have urged her to resign.
Amid mounting pressure for her to resign her seat, Ms Ferrier has held on and the Metropolitan Police has now said in a statement that regulations under which Ms Ferrier could have been punished were not in effect when she travelled.
For a by-election to take place, the MP would need to quit or be forced out through a recall petition.
This would require her to be suspended from the House of Commons for at least 10 sitting days, or at least 14 days if sitting days are not specified.
A by-election could then be forced if 10% of her constituents sign a petition.
Scottish Labour leader Richard Leonard has now said his party will select a candidate “within weeks” in preparation for a contest.
Labour held the Rutherglen and Hamilton West seat before Ms Ferrier defeated incumbent Ged Killen in last year’s general election
Mr Leonard said: “In a matter of weeks, Labour will have a candidate ready to contest a by-election in Rutherglen and Hamilton West.
“Margaret Ferrier has let down the people of this constituency and has forfeited her right to represent them in Parliament by flouting the laws on Covid that we all have to follow.
“Rutherglen and Hamilton West deserves an MP whose priority is to fight to protect workers’ jobs and incomes, that are facing such a severe threat from Covid, rather than focusing on preserving a parliamentary salary and Westminster perks.”
He added voters “deserve better than a lame duck MP, who only speaks up for her own interests”.