keep eyes open

keep (one's) eye(s) open (for someone or something)

To remain vigilant or carefully watchful (for something or someone). They should be arriving any minute, so keep your eyes open. Keep your eye open for a birthday gift we could give your mother. Tell everyone to keep their eyes open for the health inspector. We heard she'll be doing a surprise inspection someday soon.
See also: keep, open, someone
Farlex Dictionary of Idioms.

keep one's eyes open (for someone or something)

 and keep one's eyes peeled (for someone or something)
Fig. to remain alert and watchful for someone or something. (The entry with peeled is informal. Peel refers to moving the eyelids back. See also keep an eye out (for someone or something).) I'm keeping my eyes open for a sale on winter coats. Please keep your eyes peeled for Mary. She's due to arrive here any time. Okay. I'll keep my eyes open.
See also: eye, keep, open
McGraw-Hill Dictionary of American Idioms and Phrasal Verbs.
See also:
  • keep (one's) eye(s) open (for someone or something)
  • keep eyes
  • keep one's eyes open
  • keep eye out
  • keep (one's) eye(s) out (for something or someone)
  • keep (one's) eye(s) peeled (for something or someone)
  • keep (one's) eye(s) skinned (for someone or something)
  • keep eyes peeled
  • keep an eye out for
  • keep an eye out for (someone or something)