Keep a close eye/watch on somebody/something
Definition 1 of 1
Pro Tip 1/3Surveillance Metaphor
The phrase implies careful monitoring, as if keeping something within sight at all times. 
The security guard kept a close watch on the entrance during the event.
Pro Tip 2/3Attention to Detail
It suggests focusing on small details or changes to prevent problems or notice opportunities. 
She keeps a close eye on her investments to catch any market changes quickly.
Pro Tip 3/3Not Literal
It's often used figuratively. You don't need to literally watch something, but to stay informed about it. 
I'll keep a close eye on the weather forecast for our trip.