Senior Conservative figures attacked the prime minister after the party confirmed that following a series of complaints, Mr Johnson is to face a full investigation.
Some Brexiteer MPs claimed the complaints were politically driven by those who disagree with Mr Johnson over the EU, while others warned that pursuing him would tear the party apart.
One backbencher told The Independent the handling of the affair by Ms May and other party leaders had been so poor that it has increased the chances of a leadership challenge over the summer.
It came as Mr Johnson was admonished by the ministerial watchdog for signing a contract with the Daily Telegraph days after leaving office, without seeking advice on whether it could lead to conflicts of interest.
The internal Tory process looking at Mr Johnson’s comparison of Muslim women in face coverings with “bank robbers” or “letter boxes” was automatically triggered on Thursday after party HQ received several complaints.
At least one is believed to have been lodged by Conservative Muslim Forum founder Lord Sheikh, who has called for the party to “remove the whip” from Mr Johnson, meaning he would no longer represent the Conservatives in parliament.
A party spokesman said only: “The code of conduct process is strictly confidential.”
The Independent understands that following the complaints, party chiefs must now decide whether the allegation is serious enough to establish an investigatory panel to ascertain whether Mr Johnson has broken the party’s code of conduct.
Normally, the panel of at least three people – with one member independent of the party – would be appointed by Conservative chairman Brandon Lewis. One member would also be selected by chairman of the backbench 1922 committee, Sir Graham Brady.
But with anger from the Brexitwing of the party over even starting the process, party HQ confirmed that Mr Lewis – who has already criticised Mr Johnson’s comments – would not pick the panel.
If he is found to have breached the party’s code of conduct, which includes a requirement for members to “lead by example to encourage and foster respect and tolerance”, Mr Johnson could face suspension or expulsion, though it is thought unlikely.
The prime minister, Mr Lewis and a string of other ministers have already said Mr Johnson should apologise for the comments, but he has steadfastly refused to.
As the investigation process was launched on Thursday, North East Somerset MP Jacob Rees Mogg said: “It’s hard to see what he should apologise for.
“He has defended people’s right to wear the burqa whilst saying it is an inelegant garment. Neither of those two proposals are unreasonable.”
Former defence minister Sir Gerald Howarth said Mr Johnson did not go far enough, telling LBC radio: “The veil is alien to our culture. It obliterates the face and that is something that we do not accept.
“This is the United Kingdom, it’s not Saudi Arabia. We are a Christian country, we do not wear this kind of thing in public.”
He added: “It should not be tolerated and this witch-hunt against Boris is absolutely outrageous.”
North West Leicestershire MP Andrew Bridgen said: “This is politically motivated. I believe there has to be a preliminary investigation because a complaint has been made.
“But if a decision is taken to go forward to a full investigation it will prove deeply divisive for the party.”
Another Tory MP complained that the party could have allowed the story to die down, following the prime minister’s demands for an apology, but had actually now made it into something bigger.
The backbencher said: “Downing Street has got completely on the wrong side of public opinion again.
“They have a terrible track record for it – the election, the manifesto, the Chequers deal [on Brexit negotiations], the fact now that they have demanded an apology for this.
“After Chequers the membership are in a hostile mood like I’ve never seen never them before, and there are MPs saying this could all precipitate a leadership challenge in the summer and if not then at conference.”
As well as the complaints made to the Conservatives, Mr Johnson has also been reported to the police and to the Equalities and Human Rights Commission.
Cressida Dick, the Metropolitan Police commissioner, said the former foreign secretary had not committed a criminal offence.