The row was witnessed by an Evening Standard reporter, who happened to be walking past at the time of the incident.
Nathalie Pulford, 42, was dining in London's affluent Chelsea neighbourhood when she claims she spotted Mr Bercow bump her car during a parking manoeuvre.
However, Mr Bercow denies touching the car, much less scratching it as she alleges.
Ms Pulford told the Standard: "I said to him: 'Don't you realise you absolutely bashed into my car?' - he had walloped these cars - and he said 'No I didn't. It wasn't me'.
"He would not even approach the subject. He was being really haughty, being a completely arrogant toff."
Ms Pulford said: "I could tell he bumped the car behind him because it moved. And then he bumped my car as well. I only had that car re-sprayed last year and I know every scratch on it and the scratches there now weren't there before. But that's not the point, it's the fact he wouldn't even admit responsibility and apologise whatsoever. He was just very rude about it and talked to me like I was nothing."
"She was clearly very irate, but I totally reject any suggestion the scratches on her car were inflicted by me," he told the reporter at the scene.
"Am I too arrogant to say sorry? No. I'd be happy to 'fess [sic] up, but the fact is I didn't touch her car. I'm not the best driver... yes it was a tight spot, but if I had touched her car I would have felt it."
Ms Pulford was having dinner at Gaucho in Sloane Avenue with her father, who had just flown in from Texas, and her two sons, the newspaper reports.
She said: "I was sitting talking to my dad at the dinner table and just saw this car moving and said
'That car just bumped that other car. I'd better watch because my car is the other side'."
She added: "It sounds petty but it's the principle. As far as I'm concerned he's a little weasel who should take responsibility, but he's just like the rest of them."
Ms Pulford admitted to losing her cool, saying Mr Bercow "was so dismissive of me". She insisted that if he had apologised, she would "not have minded in the least".
The Evening Standard adds that a waitress at Gaucho backed Ms Pulford's version of events. Nadia Green is quoted as saying: "I saw the bump. I was taking an order and the (Bercow's) car literally touched both of the other cars to get into the space. He was trying to squeeze in."
A spokesman for Mr Bercow said today: "He strongly denies that his car hit this lady's car. If she wants to raise it with her insurer, he would be more than happy to defend himself and explain what happened.
"But he cannot understand why she thinks he hit her car. He was simply bewildered at the time."
Mr Bercow was on his way to a school charity auction in a nearby church at the time of the incident.
The police confirmed the matter had been reported.
A spokesman for the Met said: "Police were called at shortly after 01.00hrs on Thursday 3 October by a motorist reporting a damage only collision that occurred at around 18.20hrs on Wednesday, 2 October in Ixworth Place, SW3.
"The motorist has been asked to attend a police station when convenient to fully report the matter."