Avodah Zara 10a-b discusses the friendship between Antoninus (אנטונינוס), who seems to be the Roman emperor, and Rabbi Yehuda Hanasi. The gemara even describes how Antoninus built a tunnel from his home to R Yehuda Hanasi's home.
Let's assume, for the sake of argument, that this has at least some historical basis, and look at the attempts to identify Antoninus. The most obvious candidate would be Antoninus Pius, emperor from 138 to 161 CE. Soncino reports that that was the view of someone named "S.J. Rappaport". Does anyone know who that is? Here is the Soncino comment.
Wikipedia's list of his children, though Gira MIGHT be Annia Galeria Faustina Minor. The Soncino also cites Rappaport's (IMO fanciful) theory about the identity of Asvirus:
"Asverus is his adopted son Marcus Aurelius (161-180), who was also called Annius Verus — here contracted into A-S-Verus"
However, a friendship between Antonius Pius and R Yehuda Hanasi is difficult to posit, given that R Yehuda Hanasi was born in 138 CE and Antoninus Pius dies in 161, when R Yehuda Hanasi was only 22 or 23 years old. Was he such a renowned and wise figure at such a young age that the emperor would consult with him so heavily? And would he have been spending significant time in Rome so early in his life?
Another suggestion, which I *think* comes from R Mattis Kantor (the footnotes are confusing) is that Antoninus is Marcus Aurelius ("Marcus Aurelius Antoninus Augustus"), whose reign was 161 to 180, at which point RY Hanasi would have been in his early 40's, certainly old enough to have been a famous rabbi and an advisor to an emperor, and also would have spent enough time visiting Rome to get to know Marcus Aurelius.
The identification of Marcus Aurelius would fit really neatly with a bow if this story in the gemara on Avodah Zara 10a was about his son Commodus instead of his son "Asvirus". The gemara there says that Roman kings do not appoint their own sons as king, but that Antoninus did so by a special request, and his natural son Asvirus succeeded him.
In fact, Marcus Aurelius did name his own, natural born son as his successor, as opposed to the longstanding practice of adopting an heir. That succession was quite controversial back in 180 AD. Unfortunately for our identification here, that son was Commodus, not Marcus Annius Verus. Still, things could easily have gotten garbled by the time this was written down, and there were an awful lot of Imperials named "Verus" or "Severus".