Mr Johnson, who is also a Conservative MP, was involved in an altercation at traffic lights in the footage obtained by the Sun.
Speaking to BBC political correspondent Ross Hawkins, Johnson said 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 reports BBC.
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.”