'keep an eye on' definitions:

Definition of 'keep an eye on'

From: WordNet
verb
Follow with the eyes or the mind; "Keep an eye on the baby, please!"; "The world is watching Sarajevo"; "She followed the men with the binoculars" [syn: watch, observe, follow, watch over, keep an eye on]

Synonyms of 'keep an eye on'

From: Moby Thesaurus