The Conservative leader made the comments ahead of meetings with European Commission chief Jean-Claude Juncker and EU negotiator Michel Barnier in Luxembourg on Monday, AFP reported.
In an interview with the Mail on Sunday newspaper, Johnson said he was "very confident" of getting a divorce deal at an EU summit on October 17, in time for Brexit on October 31.
In an odd analogy, he compared Britain to the comic book character Hulk.
"The madder Hulk gets, the stronger Hulk gets and he always escaped, no matter how tightly bound in he seemed to be – and that is the case for this country," he said.
But Johnson is facing opposition in Parliament to his threat to leave the EU without a deal next month if his negotiations fail.
On Saturday night, one of his Conservative MPs defected to the pro-European Liberal Democrats.
Former universities minister Sam Gyimah, who has called for a re-run of the 2016 Brexit referendum, condemned Johnson for "veering towards populism".
Meanwhile former prime minister David Cameron, who led the failed campaign to stay in the EU in 2016, accused Johnson in his memoirs of only pursuing Brexit out of political ambition.
The turmoil comes ahead of another crucial week, as the Supreme Court considers the legality of Johnson's decision to suspend Parliament earlier this month until October 14.
Johnson was a leading "Leave" campaigner in the referendum, and took office in July promising to deliver Brexit no matter what.
He wants to renegotiate the terms struck by his predecessor Theresa May, which MPs have rejected, but EU leaders have so far refused.
He told the Mail there was a "very good conversation" going on about the key sticking point, the issue of the Irish border.
"A huge amount of progress is being made," he said.
Much of the opposition in London to the current Brexit deal is focused on the so-called backstop plan to keep open the Irish border.
It could keep Britain indefinitely tied to EU trade rules to avoid frontier checks between British Northern and EU member Ireland.
Brexit minister Steve Barclay insisted "the backstop needs to go", but fueled speculation that an alternative might be taking shape.
"We can see a landing zone in terms of a future deal but there is significant work still to do," he told Sky News television on Sunday.
Johnson repeated that "under no circumstances" would he delay Brexit.
However, he may have no choice after MPs last week passed a law to postpone until January if he does not get a divorce deal by the EU summit.