one eye on

one eye on (someone or something)

Partial or intermittent attention given to someone or something. She shoveled fuel into the boiler with one eye on the pressure gauge. You cannot just keep one eye on the kids. The moment you're not paying attention is all it takes for one of them to get hurt.
See also: eye, on, one

*one eye on someone or something

Fig. [get/have] a small amount of attention devoted to someone or something. (*Typically: keep ~; have ~.) I have to have one eye on you at all times or you will get into trouble. I have one eye on the speedometer of my car whenever I am driving.
See also: eye, on, one

one eye on

Paying some but not full attention to, as in He ran the rehearsal with one eye on the clock. Also see have one's eye on; out of the corner of one's eye; see with half an eye.
See also: eye, on, one