1
general::
Phrase(s): on the point of doing something [and] at the point of doing something
Fig. ready to start doing something. • I was just on the point of going out the door. • We were almost at the point of buying a new car.
McGrawhill's American Idioms And Phrasal Verbs
2
general::
Also, at the point of. On the verge of, close to, as in I was on the point of leaving when the phone rang. The first term dates from the late 1200s, the second from the first half of the 1500s.
American Heritage Idioms