Asked if Mr Johnson should say sorry, or if he should lose the party whip, Mrs May told reporters: "I am very clear, and the Government is very clear about our position on the burka, which is that women should be able to choose how to dress".
Johnson said he opposed banning burkas, but wrote that it was "absolutely ridiculous that people should choose to go around looking like letter boxes".
Johnson's comments were widely criticised by the Muslim community and MPs.
"It is ridiculous that these views are being attacked - we must not fall into the trap of shutting down the debate on hard issues", said the source.
The PM also said: "These are issues that are openly discussed but in having that discussion it's important that people think carefully about the language that they use, certain terms in describing people's appearance that Boris Johnson used have clearly caused offence, that's why I have said I agree with Brandon Lewis".
'It is ridiculous that these views are being attacked, ' the source said, adding: 'If we fail to speak up for liberal values then we are simply yielding ground to reactionaries and extremists'.
But Johnson received support from some quarters, with Conservative MP Nadine Dorries saying he "did not go far enough".
But the standoff continued even after Theresa May added her voice to calls for him to apologise - although she swerved questions about whether his comments amounted to Islamophobia.
But The Conservative Woman co-editor Laura Perrins, appearing alongside Ms Awan-Scully, claimed Mr Johnson's remarks were taken out of context.
Illustrative photo of Muslim women wearing burqas.
Tory peer Sayeeda Hussain Warsi, a former party chair and diplomat, accused Johnson of adopting the "dog-whistle" tactics of right-wing firebrand Steve Bannon, US President Donald Trump's former top aide.
Reacting to news the first person has been fined for wearing a full-face veil in public in Denmark, Mr Johnson warned against a similar "heavy-handed" movement in the UK.
"The party chairman, the Prime Minister has the right to take the whip ... that's the thing I'd like to see".
Muslim women are having their burkas pulled off by thugs in our streets & Boris Johnson's response is to mock them for "looking like letter boxes".
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".
Mr Johnson has long campaigned for equal treatment for women in developing countries, especially the provision of education for girls.
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