keep (a) close watch on (someone or something)
To monitor someone or something closely. Keep a close watch on this pot so that the soup doesn't boil over.
keep a close watch on someone or something
to watch someone or something very carefully. Let's keep a close watch on Fred and his friends. I want to keep a close watch on the house across the street.
keep a close ˈeye/ˈwatch on somebody/something
watch somebody/something carefully: Over the next few months we will keep a close eye on sales.