Friends of Mr Johnson said they were disappointed he had been recorded.
In the recording, obtained by Buzzfeed, Mr Johnson warns the UK could remain “locked in orbit around the EU” and claimed the Irish border issue – one of the main sticking points in talks with Brussels – had been allowed to dictate “the whole of our agenda”.
“It’s so small and there are so few firms that actually use that border regularly, it’s just beyond belief that we’re allowing the tail to wag the dog in this way,” he said.
The foreign secretary was apparently speaking to around 20 people in a private room after an Institute of Directors reception on Wednesday night.
It follows a day of wrangling over the government’s “backstop” plan in the event of no customs deal being agreed before Brexit.
Theresa May was forced to agree to a cut-off date of December 2021 for any interim arrangements after Brexit Secretary David Davis threatened to resign.
But speaking to reporters on route to the G7 summit in Canada, she twice refused to give a “cast-iron guarantee” that the end date would not slip beyond that.
The EU’s chief negotiator Michel Barnier will officially respond to the UK’s proposals for the first time at a press conference in Brussels later, having said on Twitter they will need to satisfy a number of tests.
In the leaked comments, Mr Johnson said the prime minister was “going to go into a phase where we are much more combative with Brussels”.
He 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.”
Brexit will be “irreversible” and will happen, Mr Johnson said, but the “risk is that it will not be the one we want”.
He added: “Unless you make the change, unless you have the guts to go for the independent policy, you’re never going to get the economic benefits of Brexit. You’ll never get the political benefits of Brexit.”
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”.
Speaking about the Treasury, he added: “They don’t want any disruption of the economy. So they’re sacrificing all the medium and long-term gains out of fear of short-term disruption.”
By Chris Mason, BBC political correspondent
The sentiments expressed might be unsurprising, but in colourful language and with devastating timing, Boris Johnson becomes the second cabinet minister in 24 hours to throw the prime minister’s authority into the pestle and mortar and pummel away.
His friends say he’s disappointed the remarks were covertly recorded.
But it’s not surprising they were.
So first there was the Brexit Secretary hokey cokey: David Davis saying it is my way or the stairway (to the backbenches), and doing so in a way where, depending whether it was 10:00 or 11:00, he was either in the cabinet or about to be out.
And now this. Johnsonian expression, including obligatory flourishes in Latin, which amounts to the same thing: a flexing of Brexiteer muscle in the direction of a prime minister desperate to hold her cabinet, her party and the country together.
It’s one heck of a job.
Mr Johnson was the keynote speaker at Conservative Way Forward’s summer reception and took questions for more than an hour – with other Conservative MPs, including Priti Patel and Conor Burns, present – according to Buzzfeed.
Asked about Donald Trump, he said he was “increasingly admiring” of the US President, who is preparing for an historic summit with North Korea’s leader Kim Jong-un.
Mr Johnson was reported to have said: “I have become more and more convinced that there is method in his madness.”
“Imagine Trump doing Brexit,” 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.”
Mr Johnson’s other comments included: