"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!"