Sayyiduna Ali (may Allah be pleased with him) says: I regret that I did not ask the Prophet (peace and blessings be upon him) to appoint Sayyiduna al-Hasan (may Allah be pleased with him) and Sayyiduna al-Husayn (may Allah be pleased with him) as mu'adhdhins.
Al-Tabarani recorded it in al-Awsat; its chain contains a narrator, al-Harith, and he is da'if (weak).
Hadith Referenceمجمع الزوائد ومنبع الفوائد / كتاب الصلاة / 1836