Have One's Eye On Meaning

(idiomatic) to seek, pursue

Example: 2011 December 14, Angelique Chrisafis, “Rachida Dati accuses French PM of sexism and elitism”[1], Guardian:
  She now wants to be a Paris MP. But Fillon wants the same Paris seat when he steps down, after next year's election. Both have their eye on the main prize: running for Paris mayor in 2014.