What is the synonym of "spy"?

Maybe the word "snoop" could be a good synonym.

I caught him snooping around in my office.

Reporters were snooping on the celebrities

You can also use "notice", "observe", "monitor", "track", "keep an eye on", "keep track of", "keep tabs on" -- all these have slightly different meanings, depending on what you mean to convey when you'd use spy.

• Use notice if you mean to say that you suddenly noticed a detail of some sort.
• Use observe if you mean to say that you're dispassionately watching something unfold, or noticing a detail or series of details with emotional detachment.
• Use monitor if you mean to say that you're continually watching something with some measurement in mind that would prompt you to step in or take action if the measurement goes outside a certain range.
• All of the keep... phrases are informal ways of saying that you're going to spend some casual effort going forward to (essentially) monitor something -- they're mostly casual ways of using the monitor sense of spy.

Notice that, strictly speaking, spy can carry a connotation of secret, covert behaviour. When you spy you have an agenda of some sort, or a secret purpose. The most neutral version of spy is to suddenly notice something: the longer the activity spent observing, the more you convey a sense of secretness.

To spy, in the sense of watching someone or something, could also mean keeping that thing under surveillance.

In that sense, spy (the verb) has approximately the same meaning as surveil.

If you mean to spy then you can use snoop (informal), keep under surveillance/observation, keep a watch on, keep an eye on, observe, spot.
If you mean spy used as noun, then you can use secret agent, undercover agent, intelligence agent, double agent, mole, snoop.

Which word/phrase suits better depends on the context.

infiltrator, undercover agent