It appears evident from the record of Paul's travels in Acts 16ff that when Paul wrote "Romans" he had never visited Rome yet:
NIV Romans 1:10
...in my prayers at all times; and I pray that now at last by God’s will the way may be opened for me to come to you.
Catholic tradition says that Peter established the assembly but there does not appear to be any evidence to that effect. In fact, Peter is not even mentioned in chapter 16 where Paul sends his various personal greetings:
And what did the Romans know of the gospel of the grace of God that Paul expounds in the letter? Was this rehashing info they already had or was he bringing to them his "faith apart from religious activities" gospel for the first time?
The most useful suggestion I have found so far is from a post on another forum:
I would suggest that the Church in Rome was planted by those "visitors" from Rome who were in Jerusalem on the Day of Pentecost and were witnesses to the outpouring of the Holy Spirit upon the apostles and those who were gathered with them. Reading Acts 2:7-11 we can note that verse 10 includes the words "... strangers from Rome ..." The NASB is a little more accurate with the words " ... and visitors from Rome ..." The Greek word literally refers to people from Rome (and the other named places) who were temporarily in Jerusalem. (Obviously for the Feast of Shavu'ot, which we call Pentecost, which was one of three annual festivals that required Hebrew men to travel to Jerusalem.)
Therefore I would suggest that the Church in Rome was the result of those Roman visitors who accepted Christ as their Savior—being either among the 3,000 men that day or the additional 5,000 a few days later—who then returned to Rome and brought with them the gospel which they shared in obedience to Christ. No doubt evangelists went to Rome from the early Church in Jerusalem, as well. Possibly even an apostle. But when Paul ended his letter to the Church at Rome (written from Corinth during the winter of 57-58 A.D.) he named over two dozen individual Christians in Rome and not one of the apostles was mentioned. Doesn't it seem odd that if Peter were there that Paul would not have greeted him by name. Read the list of names for yourself in Romans 16:1-16. Yet, perhaps there is a clue as to an evangelist who was there: Romans 16:3 says "Greet Prisca and Aquila ..." You may recall from Acts 18 that Aquila and his wife Priscilla were evangelists who traveled with Paul, and from 1 Corinthians and 2 Timothy we learn that they moved around the known world quite a bit in the service of the gospel. Prisca is a shortened form of Priscilla. Therefore these two evangelists were in Rome in 57 A.D., roughly 24 years after the Church began in Jerusalem. Perhaps they were responsible for planting it; but I doubt it. More likely they were there to teach the people the Word of God as it applied to Christ and His followers and thus give them the doctrine of the New Testament. After all, that they were teachers is clear from Acts 18: "Now a Jew named Apollos, an Alexandrian by birth, an eloquent man, came to Ephesus; and he was mighty in the Scriptures. This man had been instructed in the way of the Lord; and being fervent in spirit, he was speaking and teaching accurately the things concerning Jesus, being acquainted only with the baptism of John; and he began to speak out boldly in the synagogue. But when Priscilla and Aquila heard him, they took him aside and explained to him the way of God more accurately. —Acts 18:24-26 NASB
However, as I point out in my response to that post his use of contemporary, unscriptural language conceals a problem with that view:
Therefore I would suggest that the Church in Rome was the result of those Roman visitors who accepted Christ as their Savior—being either among the 3,000 men that day or the additional 5,000 a few days later—who then returned to Rome and brought with them the gospel which they shared in obedience to Christ.
If they traveled to Rome for Pentecost then they would be Jews rather than gentiles and they would not have heard from Peter the gospel of the grace of God that Paul taught. When Paul encounters Peter, Peter would not even sit at table with a gentile! He is not mentioned ever in association with the assembly at Rome.
So where did the assembly at Rome come from? Were they Pauline gentile believers or a Jewish fellowship? What did they believe prior to receiving Paul?
For Luke's account of Paul's subsequent entrance into Rome (which adds an important dimension for understanding what was going on) see: https://biblehub.com/context/acts/28-11.htm