Since a bizarre incident in which Rudolph W. Giuliani verbally abused a ferret enthusiast on the radio a decade and a half ago, New York City’s ferret owners have been dealing with two major problems.

One: They are, technically, criminals.

Two: Everyone thinks they are insane.

“People see it as a dumb, silly issue: ‘Who cares, weasels?’ ” said Veronica F., whose two clandestine ferrets, Watson and Nacho, live with her in Queens. “There’s a notion that we all have some sort of mental problem.”

Now, ferret owners have a chance to change the law — and maybe remedy their reputation as well.

Last spring, perhaps trying to demonstrate the relative warmth and fuzziness of the current mayor, or perhaps simply pandering to the all-important small-mammal lobby, Mayor Bill de Blasio’s administration proposed removing ferrets from the city’s banned-pet list, a zoological rogues’ gallery that also includes rhinoceroses, bats and poisonous centipedes. (A public hearing on the proposal before the city’s Board of Health is scheduled for Jan. 21.)

The possible policy shift sent a frisson of excitement through the city’s underground ferret community.