"Slavic" is a designation of a linguistic group, not an archaeological culture, and it is notoriously difficult to match these two up in this period. Absent linguistic documentation, which we don't have in this context, it is impossible to know for certain whether a particular group is Slavic or not. That being said, there was an "Amber Road" (similar to the silk road) down which amber traveled from the Baltic to the Mediterranean. It is entirely possible that it passed through territories inhabited by Slavic speakers.

According to Peter Heather’s Empires and Barbarians: the Fall of Rome and the Birth of Europe—which is basically a survey of modern scholarship on the rise of the Slavs and Germans—the current consensus is that the Slavs only appeared as a new group among the older Baltic-speaking peoples toward the end of the western empire—perhaps as a result of the disruptions following the collapse of the Hunnic empire. They are first mentioned by name in the 6th century, as an ethnic group with no political organization; during the subsequent migrations of the Avars, Bulgars, and Magyars, they are increasingly encountered by the Byzantines as refugees and captured slaves.

I think the elephant in the room concerning this question is that the Slavs had a great deal of interaction with the Romans of the Eastern Roman Empire, a.k.a. the Byzantine Empire, through trade on the Black Sea and the Slavic river systems. From the Byzantines, Slavic peoples attained their written language system, Cyrillic, and their religion, Eastern Orthodoxy, from this interaction with the Byzantines, who for all intents and purposes thought of themselves as Romans, so there was a great deal of Roman influence on the Slavs. After the fall of Constantinople, Russia even took it so far as to call its capital, Moscow, the third Rome, defender of the faith after the fall of the first and second Romes. I'm sure you were referring to the Western Roman Empire in this question, but I still wanted to point out that most of Slavic culture is based around a Roman, or at least pseudo-roman frame.