Make haste, and come to me before winter. Eubulus and Pudens and Linus and Claudia and all the brethren send thee their greeting.
Irenaus, the historian closest to the events, is quoted in Against Heresies volume 3, chapter 3, article 3 as saying,
After the Holy Apostles (Peter and Paul) had founded and set the Church in order (in Rome) they gave over the exercise of the episcopal office to Linus. The same Linus is mentioned by St. Paul in his Epistle to Timothy. His successor was Anacletus.Against Heresies was written approximately 175-185 CE.
Eusebius, in Church History Volume 3, Chapter 4 writes this about Linus:
As to the rest of his followers, Paul testifies that Crescens was sent to Gaul; but Linus, whom he mentions in the Second Epistle to Timothy 2 Timothy 4:21 as his companion at Rome, was Peter's successor in the episcopate of the church there, as has already been shown.And in chapter 13 of that same volume, he writes:
After Vespasian had reigned ten years Titus, his son, succeeded him. In the second year of his reign, Linus, who had been bishop of the church of Rome for twelve years, delivered his office to Anencletus. But Titus was succeeded by his brother Domitian after he had reigned two years and the same number of months.The additional information provided by Eusebius is quite helpful, as it gives us a very good window into the timing involved in Pope Linus's reign. He began as pope in roughly 67-68 AD, when Peter died, and continued until the second year of the reign of Titus. Vespasian died in 79 AD, which would give Linus an approximate reign of 12 years.
When tracking the passage of time from over 2000 years ago, we do come up against many obstacles but can only work with what we have available to us. Although it is scant evidence to go on, given the distance from the event it is also the best evidence that can be found until such time as we either locate a document between 2 Timothy and Against Heresies that can provide more information.