Narrated Al-Qasim bin Muhammad
`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).
Narrated `Abdullah bin `Umar
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."
Narrated Anas bin Malik
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.
Narrated Jubair bin Mut`im
A woman came to the Prophet (ﷺ) and spoke to him about something and he told her to return to him. She said, "O Allah's Messenger (ﷺ)! If I come and do not find you?" (As if she meant, "...if you die?") The Prophet said, "If you should not find me, then go to Abu Bakr."
Narrated Tariq bin Shihab
Abu Bakr said to the delegate of Buzakha. "Follow the tails of the camels till Allah shows the Caliph (successor) of His Prophet and Al-Muhajirin (emigrants) something because of which you may excuse yourselves."
Narrated Jabir bin Samura
I heard the Prophet (ﷺ) saying, "There will be twelve Muslim rulers (who will rule all the Islamic world)." He then said a sentence which I did not hear. My father said, "All of them (those rulers) will be from Quraish."