The closest I could find is:
The things which you have heard from me in the presence of many witnesses, entrust these to faithful men who will be able to teach others also. (2 Tim. 2:2)
Retain the standard of sound words which you have heard from me, in the faith and love which are in Christ Jesus. Guard, through the Holy Spirit who dwells in us, the treasure which has been entrusted to you. (2 Tim. 1:13,14)
He must hold firmly to the trustworthy message as it has been taught, so that he can encourage others by sound doctrine and refute those who oppose it. (Titus 1:9)
For everything that was written in the past was written to teach us, so that through the endurance taught in the Scriptures and the encouragement they provide we might have hope. (Romans 15:4)
I myself am convinced, my brothers and sisters, that you yourselves are full of goodness, filled with knowledge and competent to instruct one another. (Romans 15:14)
Therefore, I will always be ready to remind you of these things, even though you already know them, and have been established in the truth which is present with you. (2 Peter 1:12)
There's plenty more, but I think this demonstrates quite well that the "unbroken chain" is one of knowledge of the truth rather than some human-to-human transmission of an "office" of Apostle.
It’s sad that Catholics give up so many blessings and truths in favor of the deception of a few men.
Well, as with the claim to the “stipulated “unanimous consent” of the fathers,", it depends upon how much you can spin "unbroken" to mean absences of up to 3 years, and excluding rival popes. Under that premise, there could be no papal successors until 2,000 years later and still claim to have "unbroken" succession.
...there is no actual standard of what gap of time is acceptable, and what gap would break succession. Thus, it is simply impossible to say what gap is acceptable. For example, according to a typical list of popes (example) there was no pope during the whole years 259, 305-307, 639, 1242, 1269-1270, 1293, 1315, and 1416, not to mention the many partial years. That's over a half dozen breaks of over a year. - http://turretinfan.blogspot.com/2010/11/romes-meaningless-claim-to-unbroken.html
Roman Catholic Garry Wills, Professor of History Emeritus, Northwestern U., author of WHY I AM A CATHOLIC,
"The idea that Peter was given some special power that could be handed on to a successor runs into the problem that he had no successor...Even so, there has not been an unbroken chain of popes. Two and three claimants existed at times, and when there were three of them each excommunicating the other two, they all had to be dethroned and the Council of Carthage started the whole thing over again in 1417." (WHAT JESUS MEANT, p. 81
The Western Schism was thus at an end, after nearly forty years of disastrous life; one pope (Gregory XII) had voluntarily abdicated; another (John XXIII) had been suspended and then deposed, but had submitted in canonical form; the third claimant (Benedict XIII) was cut off from the body of the Church, "a pope without a Church, a shepherd without a flock" (Hergenröther-Kirsch). It had come about that, whichever of the three claimants of the papacy was the legitimate successor of Peter, there reigned throughout the Church a universal uncertainty and an intolerable confusion, so that saints and scholars and upright souls were to be found in all three obediences. On the principle that a doubtful pope is no pope, the Apostolic See appeared really vacant, and under the circumstances could not possibly be otherwise filled than by the action of a general council. - http://www.newadvent.org/cathen/04288a.htm