watch with an eagle eyewatch (someone/something) with an eagle eye
to watch someone or something very closely and carefully.
The teacher was watching the children with an eagle eye, making sure they behaved themselves.
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watch (someone or something) with an eagle eye
To observe very closely, without ever looking away. Like all birds of prey, eagles are known for their excellent vision. You've already been caught once, so you better believe the principal is going to be watching you with an eagle eye from now on.I've been watching that spot with an eagle eye, but still no sign of the leopard.