Vaz, 59, resigned as chair of parliament's influential home affairs select committee in September amidasex scandal over allegations involving male prostitutes. The Metropolitan Police had said it was assessing if there was evidence to open a criminal investigation.
"Following that assessment, which included obtaining early investigative advice from the Crown Prosecution Service, the Metropolitan police service is now investigating offences under the Misuse of Drugs Act 1971. The investigation started on Thursday November 10," the Met Police said in a statement.
He was also accused of encouraging others to use poppers, an illegal party drug.
The Met had received a letter soon after asking them to investigate.
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The correspondence was passed on to a specialist team, which "started an assessment process to identify what criminal offences - if any - may have been committed".
Vaz, who had apologised to his family after the newspaper allegations, said on Friday: "I welcome a full investigation as a means of establishing the true facts and I am confident the police will pursue all lines of enquiry. I will cooperate with the investigation in any way I can."
MPs voted for him to be appointed to the committee by 203 votes to seven, a majority of 196.
Electing MPs to vacancies on Commons committees is mostly uncontentious, with each party given a set allocation of places.
The rare vote on the Labour Party backed appointment ofVazcame after Conservative MP Andrew Bridgen objected to the appointment over allegations hanging overVaz.
Bridgen had said then: "An allegation in the'Sunday Mirror', with supporting video footage, implied [MrVaz] had offered to purchase class A drugs while using the services of escorts.
"A potential police investigation still hangs over him.