To approach or come close to something; to border or be on the edge of something
It was verging on dusk, and the clock had already given warning of the hour to dress for dinner, when little Adèle, who knelt by me in the drawing-room window-seat, suddenly exclaimed.
If someone or something verges on a particular state or quality, they are almost the same as that state or quality. a fury that verged on madness