The former minister Anna Soubry added: "Many One Nation Tories would not stay in the Conservative Party should Boris Johnson become leader".
Some suspected Johnson's burqa comments were meant to boost his appeal among right-wing members of the party.
Taking jibe at burqa, Johnson on Sunday mocked women who wear the full-body veil "look like letter boxes" and compared them to "bank robbers".
Sayeeda Warsi, Johnson's colleague in the Conservative Party, and a member of the House of Lords, accused him of making "hate crime more likely" with an indefensible "dog-whistle" reference to fully veiled Muslim women.
A spokesman for the party declined to confirm the investigation. "The code of conduct process is strictly confidential", the spokesperson said instead in an emailed statement.
United Kingdom former foreign minister Boris Johnson.
Furthermore, given the responses from other MPs, specifically Ms Dorries, and the broader concerns that have been raised by the Muslim Council of Britain amongst others, we believe that there must now be an independent inquiry into Islamophobia in the Party to tackle this issue once and for all.
But sources close to the Uxbridge MP made clear earlier this week that he stands by the article on Monday, in which he argued against a burka ban, but said that the garment was "oppressive" and "ridiculous".
Disciplinary action could lead to Johnson being suspended or even expelled from the Tories. By a narrow margin of 48 per cent to 45 per cent they thought he did not need to say sorry for his comments.
"With everything else going on in London to be diverting resources into an even cursory investigation into an article is weird", Mr Burns said according to The Telegraph.
Other than Armeena Khan, the former United Kingdom mayor had received responses from many other personalities including former Attorney General Dominic Grieve, Theresa May, Brandon Lewis and Ruth Davidson.
And an imam who has previously criticised the burka said Mr Johnson should not "apologise for telling the truth". "He should not have used it", May said.