Mr Johnson, who is also a Conservative MP, was involved in an altercation at traffic lights in the footage obtained by the Sun.
A source close to the mayor told BBC political correspondent Ross Hawkins it was "a colourful choice of language" .
But the source added: "He was giving as good as he got to a guy who heckles him regularly."
It is understood the incident took place over a week ago.
The general secretary of the Licensed Taxi Drivers' Association, Steve McNamara, said: "This was an incident between two grown men.
"No children were about and no-one was offended. It's a bit of a storm in the tea-cup."
Behind this incident is a dispute that has been simmering for months.
The black cab trade blame London Mayor Boris Johnson and Transport for London for failing to do enough to protect them.
Cabbies say the rise of minicab phone apps - in particular Uber - and the increase in illegal minicab touting is putting them out of business.
Cabbies are unhappy that Uber uses a mobile app to work out the cost of journeys. They argue this is the same as a taxi meter, which only black cabs are legally entitled to use in London.
There have been protests on the roads and at City Hall - and radio phone-ins are inundated with aggrieved taxi drivers.
Now the mayor is talking about putting a cap on the number of minicab drivers in London.
But the relationship between some black cabbies and Mr Johnson is looking increasingly strained.
London mayor Boris Johnson has been caught on film swearing at a taxi driver.