keep an eye on

1.  Also, keep an eye on. Look at, especially attentively or continuously; watch. For example, The teacher has his eye on the boys in the back row, or Please keep an eye on the stew. [First half of 1400s] Also see keep an eye out for.
2.  Also, have an eye to. Have as one's objective, as in We had our eyes on that birthday cake, or The Republicans have an eye to a big majority in the House. The first usage dates from the mid-1600s, the second from the early 1500s.
3.  Also, with an eye to. With a view to, regarding as an objective, as in With an eye to her inheritance, she was very attentive to her aunt. [Mid-1800s] Also see have an eye for.

The verb has one meaning:

Meaning #1: follow with the eyes or the mind
  Synonyms: watch, observe, follow, watch over