Senior Tory figures told The Daily Telegraph that they do not expect Mr Johnson to be suspended or have the whip withdrawn.
The actor, who plays funnyman Mr Bean, said that you should only apologise for a "bad joke" and that on that basis - "no apology is required".
"As a lifelong beneficiary of the freedom to make jokes about religion, I do think that Boris Johnson's joke about wearers of the burqa resembling letter boxes is a pretty good one", Atkinson wrote in a letter to the London Times newspaper, published on Friday. He said he opposed banning burqas and other face-covering garments worn by some Muslim women, but referred to a woman wearing the attire as "looking like a bank robber".
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.
The letter also said that Johnson's choice of Islamophobic words was deliberate and "was made to inflame tensions in a way that makes it easier for bigots to justify hate crime against us", adding further support to Tory peer Lord Sheik's request that Johnson lose the party whip, meaning he would no longer represent the party.
"The investigation should be thorough, impartial and objective, and carried out with sensitivity and due respect for the rights of all parties concerned", says the code of conduct. "Extraordinary to think he was foreign secretary only a few weeks ago".
"The burka and niqab are ugly tribal ninja-like garments that are pre-Islamic, non-Koranic and therefore un-Muslim".
"The code of conduct process is strictly confidential", a Conservative party spokesperson said, refusing to elaborate on the procedure. If this goes ahead, one Tory MP told The Times it would result in "civil war" in the party.
Andrew Rosindell, the MP for Romford, said that "what is now happening to Boris Johnson is a direct attack on our freedom of speech".
Mr Mitchell, who was chief whip in 2012 and worldwide development secretary before that, told BBC Two's Newsnight: "I don't think he should apologise. Nevertheless we do so because we believe it is a means to get closer to God", it added.