According to Catholicism, St. Linus was the successor of St. Peter, the Prince of the Apostle and was the second pope and bishop of Rome.
The Catholic Encyclopedia has no mention of any existing correspondence from Pope St. Linus. Any letters or other documents produced by him or to him have been lost to history. However one decree, although Apocryphal may have been his work.
All the ancient records of the Roman bishops which have been handed down to us by St. Irenaeus, Julius Africanus, St. Hippolytus, Eusebius, also the Liberian catalogue of 354, place the name of Linus directly after that of the Prince of the Apostles, St. Peter. These records are traced back to a list of the Roman bishops which existed in the time of Pope Eleutherus (about 174-189), when Irenaeus wrote his book "Adversus haereses". As opposed to this testimony, we cannot accept as more reliable Tertullian's assertion, which unquestionably places St. Clement (De praescriptione, xxxii) after the Apostle Peter, as was also done later by other Latin scholars (Jerome, Illustrious Men 15). The Roman list in Irenaeus has undoubtedly greater claims to historical authority. This author claims that Pope Linus is the Linus mentioned by St. Paul in his 2 Timothy 4:21. The passage by Irenaeus (Against Heresies III.3.3) reads:
"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."
We cannot be positive whether this identification of the pope as being the Linus mentioned in 2 Timothy 4:21 goes back to an ancient and reliable source, or originated later on account of the similarity of the name.
Linus's term of office, according to the papal lists handed down to us, lasted only twelve years. The Liberian Catalogue shows that it lasted twelve years, four months, and twelve days. The dates given in this catalogue, A.D. 56 until A.D. 67, are incorrect. Perhaps it was on account of these dates that the writers of the fourth century gave their opinion that Linus had held the position of head of the Roman community during the life of the Apostle; e.g., Rufinus in the preface to his translation of the pseudo-Clementine "Recognitiones". But this hypothesis has no historical foundation. It cannot be doubted that according to the accounts of Irenaeus concerning the Roman Church in the second century, Linus was chosen to be head of the community of Christians in Rome, after the death of the Apostle. For this reason his pontificate dates from the year of the death of the Apostles Peter and Paul, which, however, is not known for certain.
The "Liber Pontificalis" asserts that Linus's home was in Tuscany, and that his father's name was Herculanus; but we cannot discover the origin of this assertion. According to the same work on the popes, Linus is supposed to have issued a decree "in conformity with the ordinance of St. Peter", that women should have their heads covered in church. Without doubt this decree is apocryphal, and copied by the author of the "Liber Pontificalis" from the first Epistle of St. Paul to the Corinthians (11:5) and arbitrarily attributed to the first successor of the Apostle in Rome. The statement made in the same source, that Linus suffered martyrdom, cannot be proved and is improbable. For between Nero and Domitian there is no mention of any persecution of the Roman Church; and Irenaeus (1. c., III, iv, 3) from among the early Roman bishops designates only Telesphorus as a glorious martyr. - Pope St. Linus
Thus history can not prove that St. Linus is the author of any known works that exist, nor can any letters written to Pope St. Linus seem to have survived the passing of time into history. The Catholic Encyclopedia would have certainly made mention of it otherwise.