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.
He explained there was a complaints procedure within the Conservative Party and that the Party had a mechanism for dealing with these issues.
"You may not agree with the tone or the jokes, but we have a thing called freedom of speech in this country", Duncan Smith told the BBC's Today programme.
The actor, known for his comedy performances in Mr Bean and Blackadder, said the remarks were amusing.
"You should really only apologize for a bad joke. On that basis, no apology is required", he added.
Commenting on Lord Sheikh's remarks, Mr Mitchell said: "I am horrified to hear about the communications that Lord Sheikh has had".
"This is quite an important issue about free speech and it's got nothing to do with the terrible events that take place over Enoch Powell and the Rivers of Blood speech".
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The interview also toggled between light-hearted jokes and serious talks about West's mental disorder. "You got sick thoughts?" Kanye didn't have an immediate answer and sat there in thought for a few moments before Jimmy cut to commercial.
Boris was referring to plans to introduce a burka ban in Denmark when he made the remarks - which have sparked anger in some quarters.
Despite the row engulfing the Conservatives, the latest YouGov poll for The Times put the party four points ahead.
'Full bandwagon jumping territory'Conor Burns, Tory MP for Bournemouth West'It's bash Boris'Andrew Bridgen, Tory MP for North West Leicestershire " This [inquiry is] being use to try and discredit Boris Johnson from people who are anxious about him in inside the Conservative party and outside the Conservative party.
Labour, which has faced continued pressure over its handling of anti-Semitism in the party, fell three points to 35%.
But the former Cabinet minister, who is holidaying overseas, has made no response to demands for an apology.
Johnson resigned as Britain's top diplomat a month ago after claiming that Prime Minister Theresa May's proposals for a soft exit from the European Union were killing the dream of a clear, decisive split from the bloc.
Under party rules, the head of the panel may dismiss the complaints if they are found to be obviously trivial, lacking in merit or unable to be fairly investigated.