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.
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.