But without a public show of contrition, Lord Sheikh told the BBC the party should take “severe action” against Mr Johnson.
The founder of the Conservative Muslim Forum said: “Take the whip from him. Why not? He’s not a super human being, he’s a member of the party.
“The party chairman, the prime minister has the right to take the whip. That’s the thing I’d like to see.”
“I assume that more is not being said or done because of the long series of critical Brexit votes that are coming in the House of Commons and the government’s lack of a majority,” Mr Amin said.
The niqab covers a woman’s whole face apart from her eyes while the burqa adds a transparent veil over the eyes.
Mr Johnson’s comments, made in a newspaper column denouncing the burqa as oppressive for women, attracted widespread criticism. Baroness Warsi accused the Ruislip MP of trying to get “yet another leadership bid” off the ground.
Defending the former cabinet minister, Conservative MP Andrew Bridgen said there was no need to apologise.
Speaking on BBC Radio 4’s The World at One programme, he said Mr Johnson had been trying to raise the subject in a “lighthearted way”.
“I think if you can get your point across with a little bit of humour it’s very much appreciated by the public,” he added.
“Boris is seen as a clear challenger for the leadership in due course and it’s interesting the characters, Alistair Burt, love him to bits, and the party chairman, and we all know which side they are batting on.”
Mr Bridgen later told Sky News that “millions of years of human evolution” had made people seek “affirmation for our own words and actions in the faces of others and when people can’t see someone’s face it makes them feel uncomfortable”.
Additional reporting by PA
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