It is from the Sunnah to recite the tashahhud quietly.
من السنة أن يخفي التشهد.
No references are available.
Abu Dawud (986) and Tirmidhi (291) both narrated it via Yunus bin Bukayr, from Muhammad bin Ishaq, from Abdul-Rahman bin Aswad, from his father, from Abdullah bin Masud, and he related it. Hakim (1/267) also narrated it, and Bayhaqi (2/146) from him, both via Muhammad bin Ishaq, with the same wording. Tirmidhi said, "hasan gharib." Hakim said, "Sahih according to the criterion of Muslim." Muhammad bin Ishaq is a mudallis and he narrated using anaana, but he has been corroborated.Hakim (1/230) narrated it through another route, from Ala bin Abdul-Jabbar Attar, who said, Abdul-Wahid bin Ziyad told us, who said, Hasan bin Ubaydullah told us, from Abdul-Rahman bin Aswad, from his father, from Abdullah bin Masud, and he related it. Bayhaqi (2/146) also narrated it through this route. Hakim said, "Sahih according to the criterion of the two sheikhs (Bukhari and Muslim)." This is a mistake on his part, for Ala bin Abdul-Jabbar is not among the narrators of Muslim, and his teacher's teacher, Hasan bin Ubaydullah, is not among the narrators of Bukhari, though both of them are reliable, and the hadith is sahih.Tirmidhi said, following his citation of the hadith, "This is what the people of knowledge act upon." I say, I do not know of anyone who disagreed with this. Indeed, Nawawi said in Sharh Muhazzab (3/463), "The scholars are in consensus on saying the two tashahhuds quietly and on the dislike of saying them aloud, and they cite as evidence for this the hadith of Ibn Masud."