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Boris Johnson reportedly warned there may be a Brexit "meltdown" in comments made at a private dinner this week.
The foreign secretary also referred to the Treasury as the "heart of Remain", according to a report by Buzzfeed.
In recorded comments to the campaign group Conservative Way Forward he also reportedly said he was "increasingly admiring of Donald Trump".
Friends of Mr Johnson said they were disappointed remarks made at a private event had been covertly recorded.
The foreign secretary was apparently speaking to a group in a private room after an Institute of Directors reception on Wednesday night.
According to Buzzfeed, he said that the prime minister was "going to go into a phase where we are much more combative with Brussels".
He reportedly added: "You've got to face the fact there may now be a meltdown. OK? I don't want anybody to panic during the meltdown. No panic. Pro bono publico, no bloody panic. It's going to be all right in the end."
He was said to have described concerns over the border between Northern Ireland and the Irish Republic as "pure millennium bug stuff" and suggested Chancellor Philip Hammond's department was "basically the heart of Remain".
On the US president, who is preparing for an historic summit with North Korea's leader Kim Jong-un, Mr Johnson reportedly said: "I have become more and more convinced that there is method in his madness."
He added: "He'd go in bloody hard... There'd be all sorts of breakdowns, all sorts of chaos. Everyone would think he'd gone mad. But actually you might get somewhere.
"It's a very, very good thought."