1. maintain careful observation
keep an eye on him/her/them
To 'keep an eye on someone' or something means to keep it under observation or to stay informed about the person's behaviour or activities. Usually, this phrase has an element of suspicion or concern for the person's wellbeing. You will often hear parents they have to 'keep an eye on their children'. In the workplace, someone might ask you to 'keep an eye on someone' in order to monitor their progress. To 'keep your eye on' something has a similar meaning - to be attentive and aware of it: "I have my eye on the competitor." It can also mean to admire something or to wish to possess it. "I have my eye on the blue bicycle, I might buy it at the end of the month." To have your eye on something also means to monitor it closely. "I always keep an eye on spending. I don't want it to get out of hand," "Keep your eye on the clock. You don't want the dinner to burn."