one eye on

one eye on  {informal}

Watching or minding (a person or thing) while doing something else; part of your attention on. — Used after "have", "keep", or "with".

Jane had one eye on the baby as she ironed.

Bill kept one eye on his books and the other on the clock.

Chris tried to study with one eye on the TV set.

Compare: KEEP AN EYE ON.