Sayeeda Warsi, a Conservative member of the House of Lords, said Johnson was using Muslim women as a "convenient political football to try and increase his poll ratings". She said she welcomed an investigation, "but let's not pretend this is an isolated incident".
The independent panel will decide whether to refer Mr Johnson to the party's board.
If the independent panel finds him culpable, Johnson faces suspension of his membership or expulsion from the party.
Interviewed in the Daily Telegraph in July, Bannon appeared to be of a similar opinion: "Now is the moment".
Those defending Johnson have included numerous former foreign secretary's Vote Leave allies who could be expected to heartily support a leadership bid.
"It is absolutely ridiculous that people should choose to go around looking like letter boxes", he wrote.
Schools and universities should be able to take the same approach if a student "turns up. looking like a bank robber".
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The Ohio rally was Trump's third of the week and he has pledged to accelerate his pace further as the midterms approach. It was unclear Saturday why Hicks, who left the White House in March, was accompanying the president's team.
Johnson, who leaped back into contention to replace Theresa May as prime minister after resigning his post in protest against the government's Brexit plan, made the controversial comments in a newspaper column on Monday.
Mr Johnson, who is holidaying overseas, has made no response to demands for an apology.
Tousle-headed Johnson is one of Britain's best-known politicians, a popular but divisive figure known for Latin quips and verbal blunders that have included calling Papua New Guineans cannibals and accusing people in Liverpool of "wallowing" in victimhood. "It is ridiculous that these views are being attacked".
Right-wing commentator Tim Montgomerie was among those to tweet his support for Johnson, saying: "I hope Boris doesn't apologise".
"The women who wear the niqab and report into us clearly report to us they are called telephone boxes, letter boxes, bin bags, when they are abused and when they are assaulted. If we fail to speak up for liberal values then we're simply yielding ground to reactionaries & extremists".
Meanwhile, the commissioner of the Metropolitan Police said Mr Johnson's comments do not constitute a crime. Half of the biggest names in the business in this country are in prison now.