No Companion [ever] (1) said that the Prophet—may Allah bless him and grant him peace—had explicitly designated anyone other than Abū Bakr—may Allah be pleased with him—(2): neither al-‘Abbās nor ‘Alī, nor (3) anyone else. Neither al-‘Abbās nor ‘Alī—[nor anyone] (4) who loved them—claimed the caliphate for either of them or claimed that either had been explicitly designated.
Indeed, no Companion even said that anyone among Quraysh was more entitled to it than Abū Bakr, whether from Banū Hāshim or from outside Banū Hāshim (5). All of this is known (6) to scholars well versed (7) in transmitted reports, the Sunnah, and hadith; for them, it is known of necessity.
It has been reported that some members of Banū ‘Abd Manāf, such as Abū Sufyān and Khālid ibn Sa‘īd (8), wanted the caliphate to belong [only] (9) to Banū ‘Abd Manāf,