I heard in one of the lectures delivered by a member of Tableeghi Jamaat that:
a) When Malik-ul-Maut came to take over the soul of Prophet Muhammad, sallallaahu alayhi wa sallam, he stood at the door of the Prophet's house and asked for his permission (1) to enter (2) and to take over his soul.
b) And that Allaah had given a choice to the Prophet to live as long as he wished. Is there any truth in these statements? Please advise.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

All the Ahadeeth that state that the Angel of Death sought the permission of the Prophet are weak.

As regards the Ahadeeth that state that the Prophet was given the choice between life or death, then this is reported in many Saheeh (authentic) Ahadeeth. 'Aa'ishah said: “I used to hear the Prophet saying that no Prophet dies till he is given the choice between this world and the Hereafter. Thus, I heard him during the illness in which he died reciting this verse (which means): {… those will be with the ones upon whom Allaah has bestowed favor of the Prophets, the steadfast affirmers of truth, the martyrs and the righteous. And excellent are those as companions.}[Quran 4:69], then I realized that he was given that choice." [Al-Bukhari and Muslim]

Another wording of the Hadeeth reads that the Prophet, sallallaahu ‘alayhi wa sallam, used to say when he was healthy: “No prophet dies till he is shown his place in Paradise, and then he is given the option (to live or die).” [Al-Bukhari]

Allaah Knows best.