Mr Johnson said full-face veils should not be banned but looked "ridiculous".
Mr Johnson's remarks, in a Daily Telegraph article, have provoked criticism from Muslim groups, some Tory MPs and opposition parties.
Foreign Office Minister Alistair Burt told the BBC: "I would never have made such a comment, I think there is a degree of offence in that, absolutely right".
Lord Pickles called for Mr Johnson to apologise, but dismissed suggestions that is comments were comparable to Enoch Powell's Rivers of Blood speech. "If we fail to speak up for liberal values then we are simply yielding ground to reactionaries and extremists".
Among critics in the community, was chairman of the Jewish Leadership Council Jonathan Goldstein, who took to Twitter to say: "Boris Johnson's comments totally disgraceful".
Allies of Boris Johnson have said he will not apologise for comments made about the burqa which have sparked a row over Islamophobia.
Labour's Jess Phillips tweeted in response that Mr Johnson was "just a racist" in response.
He also said it made its wearers look like letter boxes or bank robbers.
After a meeting with the Scottish first minister, Nicola Sturgeon, May told reporters: "I am very clear and the government is very clear about our position on the issue of the burka, which is that women should be able to choose how to dress".
Mr Johnson said that schools and universities should expect students to remove face coverings if they turn up "looking like a bank robber".
The former Conservative party chair Sayeeda Warsi has accused Johnson of "dog-whistle" Islamophobia and criticised the lack of action by the party on the issue.
Several European countries, including France, Belgium and Denmark, have banned face-covering veils in public, but none of Britain's main political parties supported such a restriction.
"The party chairman, the Prime Minister has the right to take the whip ... that's the thing I'd like to see".
Meanwhile, Johnson's father, Stanley, told ITV's Good Morning Britain: "I think people are whipping up a little mountain out of a molehill on this one, I really do".
She told BBC Radio 4's PM programme: "Some women choose to wear it, some women are conditioned to wear it, but it's certainly out of the question for me to tell a woman what to wear".
Some suspected Mr Johnson's burka comments were meant to boost his appeal among right-wing members of the party.
- More seriously, he faced calls to resign last November amid claims his loose comments about a British-Iranian woman held in Iran had jeopardised her chances of release.