Johnson resigned as Britain's top diplomat a month ago after claiming Prime Minister Theresa May's proposals for a soft exit from the European Union were killing the dream of clear, decisive split from the bloc.
News of the investigation emerged as Mr Johnson was reprimanded for breaching rules for former ministers by failing to seek advice before taking up a job as a newspaper columnist after leaving the Cabinet.
The source said that a number of complaints had been received, and would be considered by an independent panel.
Absent from the 2016 leadership contest when key allies withdrew their support on the eve of his campaign launch, Mr Johnson failed to bring any statesmanship to the Foreign Office.
Boris Johnson is the subject of a preliminary investigation by Conservative headquarters into his comments about the burka. For, if they do remain in thrall to Mr Johnson and his blunderbus approach, the Tory party will have even less public credibility in the future.
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"Islamophobia does play a big role in Muslim women's lives", she said in a telephone interview from the Midlands. On those that wear the burka he appeared to draw comparisons with a female student who turned up to school "looking like a bank robber" - adding they should be asked to remove their face covering.
Speaking on his LBC show, he said: "I think the portrayal of Islam as a threat, the portrayal of Islam as a murderous cult is not being done by Muslims". "It is ridiculous that these views are being attacked".
"It's ridiculous these views are being attacked - we mustn't fall into the trap of shutting down the debate on hard issues", the source said. "If we fail to speak up for liberal values then we are simply yielding ground to reactionaries and extremists".
Johnson's comments drew widespread criticism and calls for him to apologise.
We understand that you have requested Mr Johnson to apologise.
There is a growing sense that "a confrontation is being forced between those in the party who think the way Johnson described women who wear a niqab was out of order and those who think it was fair comment", says The Times.