From Longman Dictionary of Contemporary Englishkeep an eye open/out (for somebody/something)keep an eye open/out (for somebody/something)WATCHto watch carefully so that you will notice when someone or something appears Keep an eye out for rabbits in the field. → eye
Examples from the Corpus
keep an eye open/out (for somebody/something)• Always keeping an eye out in case of thieves.• Though he works hard with all the kids, he keeps an eye out for the special ones.• For months, he kept an eye out.• Male speaker All you got to do is keep an eye open and watch the break lights.• Valueoriented consumers should keep an eye out for the name FabreMontmayou.• And we had to keep an eye open for police patrols.• He will keep an eye out, but he can not promise anything. 