It was said to `Umar, "Will you appoint your successor?" `Umar said, "If I appoint a Caliph (as my successor) it is true that somebody who was better than I (i.e., Abu Bakr) did so, and if I leave the matter undecided, it is true that somebody who was better than I (i.e., Allah's Messenger (ﷺ)) did so." On this, the people praised him. `Umar said, "People are of two kinds: Either one who is keen to take over the Caliphate or one who is afraid of assuming such a responsibility. I wish I could be free from its responsibility in that I would receive neither reward nor retribution I won't bear the burden of the caliphate in my death as I do in my life."
Sahih al-Bukhari Book 93, Hadith 78
← Previous Hadith `Aisha said, "O my head!" Allah's Messenger (ﷺ) said, "If that (i.e., your death) should happen while I am still alive, I would ask Allah to forgive you and would invoke Allah for you." `Aisha said, "O my life which is going to be lost! By Allah, I think that you wish for my death, and if that should happen then you would be busy enjoying the company of one of your wives in the last part of that day." The Prophet said, "But I should say, 'O my head!' I feel like calling Abu Bakr and his son and appoint (the former as my successors lest people should say something or wish for something. Allah will insist (on Abu Bakr becoming a Caliph) and the believers will prevent (anyone else from claiming the Caliphate)," or "..Allah will prevent (anyone else from claiming the Caliphate) and the believers will insist (on Abu Bakr becoming the Caliph).

Sahih al-Bukhari Book 93, Hadith 77 Next Hadith → That he heard `Umar's second speech he delivered when he sat on the pulpit on the day following the death of the Prophet (ﷺ) `Umar recited the Tashahhud while Abu Bakr was silent. `Umar said, "I wish that Allah's Messenger (ﷺ) had outlived all of us, i.e., had been the last (to die). But if Muhammad is dead, Allah nevertheless has kept the light amongst you from which you can receive the same guidance as Allah guided Muhammad with that. And Abu Bakr is the companion of Allah's Messenger (ﷺ) He is the second of the two in the cave. He is the most entitled person among the Muslims to manage your affairs. Therefore get up and swear allegiance to him." Some people had already taken the oath of allegiance to him in the shed of Bani Sa`ida but the oath of allegiance taken by the public was taken at the pulpit. I heard `Umar saying to Abu Bakr on that day. "Please ascend the pulpit," and kept on urging him till he ascended the pulpit whereupon, all the people swore allegiance to him.

Sahih al-Bukhari Book 93, Hadith 79