I recognise the modern challenges of traditional/specific dating of the NT books; and that this may suggest various answers. Are there any letters which are reliably recognised as having been written while under arrest in Rome?
Information taken from the New International Version Study Bible 2000 Edition Introductory Notes:
Ephesians: Paul may have written it about the same time as Colossians, c. A.D. 60, while he was in prison in Rome.
Philippians: It is evident that Paul wrote the letter from prison... The best evidence, however favours Rome as the place of origin, and the date as c. 61. This fits well with the account of Paul’s house arrest in Acts 28:14-31. When he wrote Philippians, he was not in the Mamertine dungeon as he was when he wrote 2 Timothy. He was in his own rented house, where for two years he was free to impart the gospel to all who came to him.
Colossians: Instead, it is to be dated during Paul’s first imprisonment in Rome, where he spent at least two years under house arrest (see Acts 28:16-31). Some have argued that Paul wrote Colossians from Ephesus or Caesarea, but most of the evidence favours Rome as the place where Paul penned all the Prison Letters (Ephesians, Colossians, Philippians and Philemon). Colossians should be dated c. A.D. 60, in the same year as Ephesians and Philemon.
Philemon: Paul wrote this short letter probably at the same time as Colossians (c. A.D. 60).
2 Timothy: After Paul’s release from prison in Rome in A.D. 62/63 (Acts 28)and after his fourth missionary journey, during which he wrote 1 Timothy and Titus, Paul was again imprisoned under Emperor Nero c. 66-67. It was during this time that he wrote 2 Timothy. In contrast to his first imprisonment, when he lived in a rented house (Acts 28:30), he now languished in a cold dungeon (2 Timothy 4:13), chained like a common criminal (2 Timothy 1:16; 2:9). His friends even had a hard time finding out where he was being kept (2 Timothy 1:17). Paul knew that his work was done and his life was nearly at an end (2 Timothy 4:6-8).