It’s possible, although I suspect it’ll be more like “being on a watch list”, which means they probably just automatically scan various databases for your name and collect any red flags that pop up. Then, after you get a certain # of red flags popping up, when they print out the report of, “These people need to be questioned”, the names are send to the local field office and the person gets a phone call / a visit.
But otherwise, a person being watched probably would never know.
This is all “probably” though. The FBI may have extra powers but they aren’t HUGELY staffed. That’s why I suspect most of the watching is data collection, which turns into reports, which make up smaller reports, until it’s part of “Possible threat – investigate further!” report which would probably have just a couple of names on it.
Then again, I could all be wrong. I’m just guessing.