The confrontation came during the Mayor's monthly phone-in show on LBC 97.3 - thought to be only the second time the two men have spoken.
The dispute over ticket office closures was deadlocked tonight with millions of commuters almost certain to be caught in the crossfire.
London Underground says services will be affected from around 9:30pm on Tuesday 4 February.
– Spokesman for David Cameron
Bob Crow's strike is plain wrong and Bob Crow should call it off rather than inflict misery on hard-working families in London.
During his radio phone-in on LBC 97.3, Boris Johnson told Bob Crow: "What you have got to do is call off this pointless strike which will do nothing but cost your members. London's Mayor told the RMT Union leader to "do the right thing for yourself, for your members for London, get round the table."
Pictures from LBC 97.3
Bob Crow has called in to Boris Johnson's LBC 97.3 phone-in saying: "All we want is to express our view. All we want is in an opportunity to talk."
Boris Johnson responds by saying: "The best way to engage is to call off the strike and I will be happy to sit down and talk... but I can't sit down and negotiate with a gun held to the head of Londoners."
RMT leader Bob Crow has revealed on live radio his "secret" plan to ambush Boris Johnson at City Hall. Crow plans to turn up uninvited at the Mayor's office with TSSA leader Manuel Cortes. The confrontation plan was hatched last night following the breakdown of peace talks at ACAS.
Further talks took place today under the auspices of ACAS in a last ditch effort to resolve the dispute over the 1,000 job cuts and the closure of all ticket offices.
Despite the best efforts of your negotiators, who have spent over seven days at Acas trying to resolve the matter, management has refused to budge.
They have refused to stop seeking volunteers during the consultation period which makes a mockery of the whole process.
– Bob Crow, RMT Union
As a result of the above, our campaign of industrial action continues with strike action commencing across the company tomorrow. I urge you to stand shoulder to shoulder with your colleagues during the strike to stop these unnecessary and politically motivated cuts to LU staff and services.
– Phil Hufton, London Underground Chief Operating Officer
We proposed a cooling-off period which would have seen the unions lift their threat of strike action while we would extend our formal consultation period and continue talks at Acas.
However, the leaderships of the RMT and TSSA continue to threaten strikes. This is despite our commitment that all Tube stations will remain staffed and controlled at all times when services are operating.
In future, there will be more staff in ticket halls and on platforms to help customers buy the right ticket, plan their journeys and to keep them safe and secure.
London Underground bosses say their latest offer to call off the strikes has been rejected by the unions. Transport for London offered a cooling off period until March 5, where it would suspend the process of making workers redundant - but only if the strike action is called off.
Bob Crow has described criticism of the holiday he took to Brazil last week as "a load of old cobblers", adding:
"I said I would be back if needed, but I have not been involved in the talks. It has made no difference whatsoever."
The RMT boss revealed his cruise holiday was advertised in the Daily Mail, the same paper which took a photo of him in his Bermuda shorts.