"To keep a close eye on something" = To observe closely. To watch closely. In English we can say: "Keep an eye on..." meaning: Watch, observe, oversee. or "Keep a close eye on..." which means: Watch carefully more carefully than usual. EX: "Keep an eye on the weather, it might change in the next few days" EX: "Keep a close eye on the weather, there might be some severe weather soon" EX: "Keep an eye on the children for me, okay? I will be right back." EX: "Keep a close eye on those juveniles. They are up to no good!"

The answers here are way too easy for native speakers to give and also way too long. Keep a close eye on something = watch over something closely.

The correct phrase is "to keep a close eye on (something)". It means to watch carefully, to not let something out of your sight. "Keep a close eye on that pot, the water is about to boil! Don't miss it!" also used for people: "Keep a close eye on your little brother. Don't let him go outside"