The latter claim was made by European Commission president Jean-Claude Juncker himself, the FT said.
One official said Mr Johnson had a “penny-dropping” moment when he was told that replacing the Irish backstop would not prevent customs checks that cross the border with Ireland.
A Number 10 official described the descriptions of what occurred at the lunch as “nonsense”.
In or out of the single market?
Despite his insistence on a “clean break” with the EU in the event of a no-deal Brexit, Mr Johnson has previously insisted he is in favour of the single market.
A clip of a speech that went viral earlier this year saw Mr Johnson state: “We will remain a paid, valued, participating member of the single market. Under no circumstances in my view will a British government adjust that position.”
Boris Johnson speaking before the referendum: "I would vote to stay in the single market. I'm in favour of the single market." pic.twitter.com/UCeifCf0dC
— Adam Bienkov (@AdamBienkov) October 2, 2016
Speaking before the EU referendum, Mr Johnson also stated: "I would vote to stay in the single market. I'm in favour of the single market.”
He added to Sky News: "I'm in favour of the single market. I want us to be able to trade freely with our European friends and partners.”
Irish backstop alternative
Britain’s Brexit negotiator, David Frost, has proposed an alternative to the Irish backstop that would see common rules for checking animals and animal products established across the whole island of Ireland.
The system of sanitary and phytosanitary (SPS) checks on agricultural products was hoped to prove acceptable to Ireland but Irish Prime Minister Leo Varadkar said they only account for around 30% of border checks.
Michel Barnier, the EU’s chief negotiator, said on Wednesday that despite the UK making clear what its preferred choice for the backstop alternative was, the two sides shouldn't "pretend to be negotiating" if there was no real progress.
He said: "That is not enough to move towards achieving a solution. We need a legally operative solution in the withdrawal agreement which fully responds to each one of the problems."
When Mr Johnson left his two-hour lunch with Mr Juncker, he was greeted with cries of "Go home Boris" and "stop Brexit" from pro-EU demonstrators.
He then headed to the Ministry of State for a meeting with Luxembourg’s PM Xavier Bettel and was again met by a noisy chorus, including chants of "bog off Boris" and "tell the truth”.
After the meeting he scrapped an appearance at a planned joint press conference with Mr Bettel because he feared being drowned out by the heckles.
Instead his lectern was left empty by Mr Bettel, who ridiculed Mr Johnson and stuck the boot in.
Mr Johnson is to defend his controversial decision to suspend Parliament for five weeks at the UK's highest court on Wednesday.
Lawyers representing the PM will argue that his advice to the Queen to prorogue Parliament was lawful, and that in any event the court cannot interfere in political matters.
The Supreme Court in London is hearing historic appeals from two separate challenges, brought in England and Scotland, to the prorogation of Parliament.
Mr Johnson says the five-week suspension is to allow the Government to set out a new legislative agenda in a Queen's Speech when MPs return to Parliament on October 14.
But those who brought the legal challenges argue the prorogation is designed to prevent parliamentary scrutiny of the UK's scheduled departure from the EU on October 31.