Have one eye/half an eye on something

Pro Tip 1/3

Refers to doing another activity with less focus while primarily engaged in something else.

He had one eye on the report while watching the football game.

Pro Tip 2/3

Suggests awareness of potential distractions that could interfere with the main task.

While studying, she kept half an eye on her notifications.

Pro Tip 3/3

Implies readiness to respond quickly to a change or need while engaged elsewhere.

I have one eye on the weather in case it turns bad during our hike.