When Agent Smith catches Neo and takes him in for "questioning," they're interested in Morpheus and plant a probe in Neo to help them find Morpheus. They don't seem at all interested in Neo for himself, but only for his ability to lead them to Morpheus. At this point the Machines are not aware of how important Neo is in the long run.
When do the machines realize just how important Neo is? Do they not realize this until Agent Smith can't kill him? Or is it at another time?
The Architect knows that eventually, someone like Neo must appear. It's kind of "built" into the system. So it's reasonable to assume that the machines keep watch (they never tire of boring tasks).
What we know is that it must happen between the time after the Oracle meets Neo and Neo meets with the Architect. My speculations:
The Oracle is (among other things) part of the security of the Matrix. We only know what she tells us, not what her complex relationship to the Matrix really is. She might have notified the rest of the system of her assessment of Neo. If that was the case, why didn't the machines kill Neo instantly?
If the Oracle assumed Neo to be curious but not dangerous, the machines might label him has "potential threat" and leave him alone until the pattern becomes more clear.
Or the Oracle might keep some thoughts to herself. Like a mother, who wants to see how far her children will go. In this case, the low level security systems (the Agents) might be aware that something is on foot but Neo might be still one file among the other people who defected to Zion.
So I can't find any evidence either way here.
The Architect knows what Neo is and why. At this time, it's safe to assume that they know what he is, since the Architect seems to be an integral part of the Matrix (i.e. he works with the Matrix while the Oracle might not share all their thoughts with the other machines).
At the end of the first movie, Neo "kills" Agent Smith (or turns him into a potent super virus).
I think it's safe to assume that some part of the Matrix realizes that Neo is special at this point. But that doesn't necessarily make Neo "important".
For example, we don't know what happens to all the other people who reach a level close to Neo's. How does the Matrix deal with them generally? Neo is hunted as in "shoot to kill" but that doesn't seem to apply for the children playing in Oracle's flat.
Before that, Neo fights with an Agent on the roof of the building and Neo manages to evade the bullets. According to Trinity, this is a very unusual ability. It's unclear whether this triggered any monitor/sentry programs - at least there was no unusual response (like flooding the area with agents or blowing up the building in a kind of ad-hoc terror bombing) inside the Matrix. This doesn't mean that the backbone processes didn't notice what was happening.
So maybe Neo's aggressive behavior triggered the Agent response while a more meditative approach would have been ignored or fostered by the Matrix.
Agents have only a limited or no awareness of the One, the prophesy or the true nature of Zion. To them, Neo is the "Anomaly". As for the machines, however (the Architect et al.) the suspicion would have grown with the following events:
The Machines (at least some of them) always knew Neo was the One.
Neo's conversation with the Architect makes this clear:
First of all, the Architect is sitting in a room with a bunch of small TVs. Remember that at the beginning of Neo's initial interrogation by the Agents in The Matrix the camera first shows small TVs -- exactly like the ones the Architect watches -- and zooms into one of them. This indicates that the Architect was watching Neo's interrogation. The only reason why the Architect himself would watch the interrogation is if he knew (or at least strongly suspected) that Neo was the One. This was before Neo was even unplugged from the Matrix so there would be no reason for the Architect to suspect that Neo was the One unless he already knew.
Second, several of the Architect's quotes provide additional evidence (emphasis added):
The function of the One is now to return to the Source, allowing a temporary dissemination of the code you carry, reinserting the prime program.
...
Your 5 predecessors were, by design, based on a similar predication - a contingent affirmation that was meant to create a profound attachment to the rest of your species, facilitating the function of the One. While the others experienced this in a very general way, your experience is far more specific - vis-a-vis love.
The One has a designed function which includes code that must be disseminated to reinsert the prime program. This code almost certainly comes from the Machines in order for the One to carry out his function, and in any case the Zionists never attributed Neo's abilities to "code" but rather to his rejection of the Matrix.
Thus, the Architect at the very least seems to have known that Neo was the One before he was even unplugged from the Matrix. Given the Oracle's role in the function of the One she almost certainly knew as well. It is not clear which other Machines and programs knew about Neo but perhaps some others knew and it's even possible that all of them knew. Even the Agents might have known -- they are programs carrying out their purpose and would have simply followed their orders to attack the One even if they knew (it's not like they were going to be deleted forever if defeated). Still, the Agents probably did not know -- they didn't need to know, and Agent Smith seemed to think he could get out of the Matrix once he acquired the access codes to the Zion mainframe and destroyed Zion.
