What interactions were there between the Greek world and Rome before the foundation of the Republic?

This is very similar to the question posted by /u/acaddgc on /r/AskHistorians a few days ago

There was a brief response posted there, but I will expand a little on what /u/MichaelJTaylorPhD said. Dr. Taylor is right that we don’t hear of Rome in early Greek sources. It’s worth stretching his point back further, however, by noting we do hear of Tyrrhenians - a Greek term possibly meaning Etruscans, though more likely Italians in a broader sense – in Hesiod (Theogony 1015). This establishes a possible eighth-century knowledge of the peoples of Italy. Likewise, Tyrsenian pirates are mentioned in the Homeric Hymn to Dionysus (though the dating on those is a bit more complicated, and they certainly came after the “Homeric” poems). Nevertheless, there is no Rome to be found here.

This was not because Rome didn’t exist, of course. The area which became the city had been inhabited since the Bronze Age, and was a significant population center throughout the Iron Age. It was not anything special, though. As far as primary centers in its environs go, Rome was large but not considerably larger than other settlements like Veii. Thus, there was nothing that would have made Greeks stand up and shout “Rome! Rome!”

On the other hand, there were relations between Greeks and Romans in the monarchical period, which preceded the Republic. Evidence of imported goods from Greece can be found throughout central Italy in this period (e.g. Cifani 2021). But how we interpret these findings is another issue all together. They were almost certainly imported by Greek merchants, though their distribution throughout the region may have been handled by locals. We know that Greeks visited ports in the area, such as at Pyrgi, one of the ports of the Etruscan city Caere. The Archaic period was one of regular interactions and exchanges between cultures in the Mediterranean (e.g. Hall 2019). It is within reason to assume that Greek traders visited Rome directly, coming up the Tiber from its mouth either on boat or on foot. Pottery evidence supports this (for instance, search the Beazley Pottery Database for Greek pots found in Rome). Greeks may have encountered Romans at sea, as well, as the old notion that Romans completely eschewed the waves before the First Punic War is no longer tenable.

Unfortunately, we cannot go into much more detail than this about Greek-Roman interactions in the years before the Republic. In Justin’s epitome of Pompeius Trogus’ history, we do find some fanciful stories about Greeks in Rome before Rome existed. Evander and Hercules were both supposedly present. But, more related to your question, the Phokaians who went on to found Massilia supposedly stopped in at Rome to say “hi” on their way from the eastern Mediterranean to the mouth of the Rhone (Justin 43.3.4). This story is not impossible, in fact it very easily could have happened, but establishing a positive and long history between Rome and Massilia was important for Pompeius Trogus, who came from the area around the latter. Thus, we need to read it carefully and probably reject it as historically true.

To address the question originally asked by /u/acaddgc about their relationship before the “power balance was in Rome’s favor,” I would encourage you to not think of “international” relations in the ancient world along the lines of modern states. Relationships between sovereign entities was complex in the Archaic period, but I don’t think it’s safe to presume an overly pragmatic approach as we would expect in the modern world. Power is relative, and we can’t really say what would make one community think of another as “more powerful” than itself in this era.

We do know that Greeks at one time or another feared pirates emanating from Italy. Strabo's (6.2.2) comment that they "were so afraid of the bands of Tyrrhenian pirates" that they avoided the region is obviously false. But, it likely echoes some sort of apprehension about staying too long in the waters off of Italy. This is perhaps the closest we get to being able to talk about how groups felt about power dynamics at the time.

I hope that this answer has added a bit of clarity. I encourage you to check out the references I made in this post, as well as the suggested reading below.

Cifani, G. 2021: The Origins of the Roman Economy: From the Iron Age to the Early Republic in a Mediterranean Perspective, Cambridge University Press.

Hall, J. 2019: "Cosmopolitan Etruria: International ties in central Italy," Ancient History 22: 16-21.