In Plutarch's Camillus, he says that Herclides Ponticus (390-310 BC) called Rome a Greek city. "For Heracleides Ponticus, who lived not long after that time, in his treatise "On the soul," says that out of the West a story prevailed, how an army of Hyperboreans had come from afar and captured a Greek city called Rome, situated somewhere on the shores of the Great Sea."

http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Lives/Camillus*.html

Edit: better source without typos.

This should about answer your question. The earliest known reference to the city was either Aristotle's or Heraclides Ponticus' mention of the sack of the city. Pliny claims that the first serious mention of the city, since both those previous ones were simply in passing, was in Theophrastus, but if so it's been lost to us--a surviving reference is made to Rome by Theophrastus, but it's a brief and almost certainly bogus mention of Rome sending 25 ships to Corsica. But as I say in the answer linked, the city of Rome had quite close connections with much of Etruria and Magna Graecia--Greek cities in the area would've been perfectly aware of the existence of the city, and we know of quite a lot of trade. But by Aristotle's lifetime the city was little more than a big village on a couple of hills far out in Italy somewhere--hardly the sort of thing that would have been especially important to an Athenian, especially when the Etruscans were so much better known

Counter question (I do not know the answer to OP's question but I am curious) - Was there ever any mention of the proto-Roman culture, i.e. The Etruscans? Or were these peoples considered Greek as well?

So, were the early Romans just the dwellers of the city of Rome? What was the influence that early Rome had on its surroundings?

When did "Rome" or "Romans" started to mean something with a larger geographic reach?

Additional question: when is the earliest reference to the myth of Aeneas founding or resettling at Latium? Was it an entirely Roman invention?