one eye on

one eye on  {informal}

Посматривать на, или приглядывать за, или размышлять о чем-л. или ком-л., занимаясь чем-л. еще; Уделять часть внимания. — Исп. после "have", "keep", или "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.

Ср.: KEEP AN EYE ON.