Acts 2: Now there were staying in Jerusalem God-fearing Jews from every nation under heaven. When they heard this sound, a crowd came together in bewilderment, because each one heard them speaking in his own language. Utterly amazed, they asked: “Are not all these men who are speaking Galileans? Then how is it that each of us hears them in his own native language? Parthians, Medes and Elamites; residents of Mesopotamia, Judea and Cappadocia, Pontus and Asia, Phrygia and Pamphylia, Egypt and the parts of Libya near Cyrene; visitors from Rome (both Jews and converts to Judaism)…. Those who accepted [Peter’s] message were baptized, and about three thousand were added to their number that day.
Here’s how Lampe assesses the possibility that there were Christians in Rome as early as shortly after Pentecost.
The beginnings of pre-Pauline Christianity in Rome are shrouded in haze. Pre-Pauline Christians are attested for Rome (Romans; Acts 28:15) and Puteoli (Acts 28:13f.) …
The Christian presence in Puteoli and Rome correlates with a twofold background. (a) Jews had lived in Puteoli since Augustan times (sources), perhaps Aquileia in the north, and Puteoli accommodated the only pre-Christian Jewish settlements in Italy known to us. This is one more confirmation that earliest Christianity spread along the routes that Judaism had already followed: the synagogues were the setting for the first Christian mission. (b) the Jewish as well as the Christian “axis” Puteoli-Rome has a particular economic-historical background. The stretch Puteoli-Rome was the main trade route between the East and the city of Rome in the first half of the first century. The road of Judaism and Christianity from the east to Rome followed in the footsteps of trade. … That Judaism and Christianity made their way to Rome through Puteoli … was typical of the entrance of eastern religions into the world’s capital city. (Lampe, pgs 7, 9-10)
Note that this is in contrast to Irenaeus, one of the earliest Christian apologists, who wrote that the church at Rome was “founded and set up by the two most glorious apostles Peter and Paul.” (Against Heresies, 3.3.2). It may be possible that Peter and Paul came later and organized what was already there, but that they “founded” the church was simply not true.
We know from Paul’s letter to the Romans that there was a well-established, well-organized church in Rome, far earlier than the time he got there. And what was there, we know from Romans 16, was a network of churches that met in people’s homes.