The story of the birth of Imam 'Ali (a) in the Ka'ba is cited in three ways in different sources:
In the book, Manaqib by Ibn al-Maghazili (d. 483/1090-1), Imam al-Sajjad (a) is quoted as saying that the Prophet (s) saw that Abu Talib was sad and asked him what had saddened him. Abu Talib said that his wife, Fatima bt. Asad, was in labor pains. Abu Talib and Fatima went to the Ka'ba, together with the Prophet (s). Abu Talib made Fatima bt. Asad sit inside the Ka'ba. When 'Ali was born, the Prophet (s) took her to her house.( Ibn Maghāzīlī, Manāqib ʿAlī b. Abī Ṭālib, p. 58-59; Ibn Ḥātam, al-Durr al-naẓīm, p. 225)
According to a hadith cited in Bisharat al-Mustafa by 'Imad al-Din al-Tabari, a Shiite scholar of fifth and sixth/eleventh and twelfth centuries as quoted from Yazid b. Qa'nab, Fatima bt. Asad was near the Ka'ba when she felt labor pains. She asked God to ease the labor for her. At this time, the back part of the Ka'ba (near al-Rukn al-Yamani) was split and Fatima went inside. Then the gap was closed again. Yazid b. Qa'nab is quoted as saying: "whatever we did, we could not open the door of the Ka'ba. Thus, we understood that this was done by God. Fatima bt. Asad left the Ka'ba after four days while she had 'Ali in her arms." (Ṭabarī, Bishārat al-Muṣṭafā, vol. 1, p. 26-27)
According to al-Hasan b. 'Ali al-Tabari (alive in 701/1301-2) in his book, Tuhfat al-abrar, Fatima bt. Asad, Imam 'Ali's mother, had labor pains when she was circumambulating the Ka'ba, and she could not return home. Thus, she turned towards the Ka'ba. The door of the Ka'ba was opened, and she went inside. Then the door was closed and she gave birth to 'Ali inside the Ka'ba. She was inside for three days.  (Ṭabarī, Tuḥfat al-abrār, p. 164-165)