The foreign secretary also took a swipe at Chancellor Philip Hammond, calling the Treasury "the heart of Remain", in comments to a private dinner.
He said the Brexit talks were heading for "meltdown" and Leave supporters may not get the deal they expected.
Downing Street said Theresa May had "full confidence" in Mr Johnson.
Friends of the foreign secretary said they were disappointed he had been recorded, as one Labour MP attacked Mr Johnson's "Trump tribute act".
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.
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