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.
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.