The most common and most likely answer to this crossword clue is the 9 letter word SUPERVISE
I've seen this crossword clue in The Guardian
S
U
P
E
R
V
I
S
E
Buddy explains!

The straight part
We know that "Keep an eye on" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case keep an eye on = supervise.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SUPERVISE mean?
SUPERVISE verb- To oversee or direct a task or organization.
- To look over so as to read; to peruse.