keep an eye on, to

keep an eye on someone or something

 and have an eye on someone or something; keep one's eye(s) on someone or something; have one's eye on someone or something
Fig. to watch someone or something; to monitor someone or something closely. I have my eye on the apple tree. When the apples ripen, I'll harvest them. Please keep an eye on the baby. Will you please keep your eye on my house while I'm on vacation?
See also: eye, keep, on
McGraw-Hill Dictionary of American Idioms and Phrasal Verbs.

keep an eye on

1. To watch over attentively; mind.
2. To watch closely or carefully: keep your eye on the ball.
See also: eye, keep, on
American Heritage® Dictionary of the English Language, Fifth Edition.

keep an eye on, to

To keep watch over something or someone. See also eye for/to the main chance; keep your eyes. . .
See also: eye, keep
The Dictionary of Clichés by Christine Ammer
See also:
  • have your eye on somebody/something
  • keep an eye open
  • keep an eye out for
  • keep an eye out for (someone or something)
  • keep (one's) eye out for (someone or something)
  • have (one's)/an eye out for (someone or something)
  • see eye to eye about (something)
  • get (one's) eye
  • get eye Go to catch
  • catch somebody's eye