on the verge of (something)
Very closely approaching something; very nearly at the point at which something will happen. After weeks of failed experiments, I think we're finally on the verge of a breakthrough.
on the verge (of doing something)
and on the verge of somethingat the very beginning of doing something; just about to do something. Bill was on the verge of leaving town when he found a job. Susan was on the verge of laughter, so she left the lecture hall.