On this day we celebrate the feast of St. Linus, who, according to tradition, was the first Pope after the Apostles.
"Irenaeus thought that the Church had been 'founded and organized at Rome by the two glorious Apostles, Peter and Paul,' and that its faith had been reliably passed down to posterity by an unbroken succession of bishops, the first of them chosen and consecrated by the Apostles themselves. He named the bishops who had succeeded the Apostles, in the process providing us with the earliest surviving list of the popes--Linus, Anacletus, Clement, Evaristus, Alexander, Sixtus, and so on down to Irenaeus' contemporary and friend Eleutherius, Bishop of Rome from AD 174 to 189."
--Saints and Sinners: A History of the Popes,  by Eamon Duffy, Yale University Press, Third Edition, 2006, page 2.
"Yet matters are not so simple. The popes trace their commission from Christ through Peter, yet for Irenaeus the authority of the Church at Rome came from its foundation by two Apostles, not by one. Peter and Paul, not Peter alone. The tradition that Peter and Paul had been put to death at the hands of Nero in Rome about the year AD 64 was universally accepted in the second century, and by the end of that century pilgrims to Rome were being shown the 'trophies' of the Apostles, their tombs or cenotaphs, Peter's on the Vatican Hill, and Paul's on the Via Ostiensis, outside the walls on the road to the coast. Yet on all this the New Testament is silent. Later legend would fill out the details of Peter's life and death in Rome . . . . But they are pious romance, not history, and the fact is that we have no reliable accounts either of Peter's later life or of the manner or place of his death. Neither Peter nor Paul founded the Church at Rome, for there were Christians in the city before either of the Apostles set foot there. Nor can we assume, as Irenaeus did, that the Apostles established there a succession of bishops to carry on their work in the city, for all the indications are that there was no single bishop at Rome for almost a century after the deaths of the Apostles. In fact, wherever we turn, the solid outlines of the Petrine succession at Rome seem to blur and dissolve." Pages 2-3.
Linus may be mentioned in Eucharistic Prayer I,  but it is optional. See page 17.
Portions of the Second Epistle to Timothy  are read on four Sundays in Year C, though never 4:21: "Make haste to come before winter. Eubulus and Pudens and Linus and Claudia and all the brethren, salute you."
Scripture scholars do not know if that Linus and Pope Linus are one and the same, but it is reasonable to think they may be. Perhaps Eubulus and Pudens and Linus and Claudia were bishops, who, like Timothy, had been mentored and consecrated by Paul.
Click here  to see the statue of St. Linus in St. Peter's Basilica.