Water cannon were routinely used to disperse crowds during the civil unrest in Northern Ireland but have never been used on mainland Britain, and the issue is contentious.
After a wave of rioting in London in August 2011, police said there could be limited use of water cannon in any future unrest, but the government is still considering the issue.
Despite the lack of official approval, Johnson has agreed the purchase of three second-hand vehicles from the German police, in a deal which he said could not wait any longer.
Pressed as to whether he would be willing to be filmed being hosed down, he said: "I'm not quite sure whether I want to stand in front of a water cannon -- I haven't done anything to deserve it."
Also Read
But after a few moments' consideration, he told the presenter: "OK. Man or mouse. Alright, you've challenged me to this. I suppose I'm going to have to do it now.
"I am doing this unfortunately from off-base, so I can see all my poor press people are going to be tearing their hair out. Never mind, it's got to be done."
Both Johnson and May are likely contenders to replace Prime Minister David Cameron as leader of their Conservative party in the event of defeat in the 2015 general election.
The mayor insisted the purchase was purely about getting a good deal and saving money, and claimed that Londoners were "pretty overwhelmingly in support of the measure".
Cameron's spokesman refused to take sides, saying the premier supports the police having water cannon in principle but thinks the correct procedures should be followed before they are put into use.