On The Point Of Meaning

(idiomatic) Very nearly; imminent; close.

Example: 1963, Margery Allingham, chapter 19, The China Governess[1]:
  As soon as Julia returned with a constable, Timothy, who was on the point of exhaustion, prepared to give over to him gratefully. The newcomer turned out to be a powerful youngster, fully trained and eager to help, and he stripped off his tunic at once.
  The old building seemed to be on the point of collapse.‎