The former foreign secretary, who resigned last month, faces claims he breached the Conservative Party’s code of conduct in an article for the Sunday Telegraph.
In it, Mr Johnson questioned why Muslim women who wear the full-face veil would “go around looking like letter boxes”.
He added he would “feel fully entitled” to ask a woman “looking like a bank robber” in a school or university lecture theatre to remove her burka.
Sky News understands several complaints about Mr Johnson’s remarks have been received by the party, automatically triggering the investigation.
If it is escalated, a panel of at least three people will be set up, appointed by party chair Brandon Lewis.
Mr Lewis previously criticised Mr Johnson on Twitter and asked him to apologise, but later refused to answer journalists’ questions.
As the investigation concerns an MP, chair of the 1922 Committee Graham Brady would also nominate a panel member to adjudicate on the complaint.
All must have “appropriate experience and no prior involvement in the complaint”.
If they find Mr Johnson has breached the code of conduct, Mr Lewis or Prime Minister Theresa May could take action including suspending or expelling him from the party.
A Conservative Party spokesperson refused to comment on the disciplinary procedure.
They only said: “The code of conduct process is strictly confidential.”
A source close to Mr Johnson said: “The world has gone mad. Or specifically the people at Matthew Parker Street [Conservative Central Office] have gone mad.
“They and Number 10 would be very, very wise to calm this down very, very quickly.”
A Labour source told Sky News: “The Tory party is on trial. Either Boris is guilty or the Tories condone Islamophobia.”
It comes after head of the Metropolitan Police, Cressida Dick, ruled out a criminal investigation into the comments.
“I spoke last night to my very experienced officers who deal with hate crime,” she told the BBC’s Asian Network.
“And although we have not yet received any allegation of such a crime, I can tell you that my preliminary view from having spoken to them is that what Mr Johnson said would not reach the bar for a criminal offence.
“He did not commit a criminal offence.”
What is a burka, niqab or hijab?
- Hijab: The word, meaning “barrier” and referring to the act of covering up in general, is used to describe headscarves worn by Muslim women. They come in different styles and colours, but those commonly worn in the UK cover the head and neck and leave the face clear.
- Niqab: The niqab is a veil covering the face which leaves an area around the eyes clear. It is worn with a headscarf, and can also be worn with a separate eye veil.
- Burka: The burka is a one-piece veil that covers the entire face and body. It has just a mesh screen for women to see through.
A Sky Data poll has since found 45% of people think Mr Johnson should apologise for his comparison of Muslim women who wear the burka. Some 48% think he should not apologise.
Mr Johnson has also been censured by a parliamentary committee that vets appointments of ministers and senior civil servants once they leave their roles.
The Advisory Committee on Business Appointments said it was “unacceptable” he failed to obtain advance clearance before resuming his Daily Telegraph column, the place he made the comments that caused controversy.