In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
Answering your question, the Fatwa Center at Islam Q and A, states:
If he had done that, he would have taught it to his companions and they would have narrated it to others. The Prophet said: “Pray as you have seen me praying.” (Al-Bukhari and Muslim)
Sheikh Ibn Qudamah said:
“Ibn Sireen and Ibn al-Mundhir said (concerning sujud as-sahw): there is tasleem [saying salam] in them but there is no tashahhud.
Ibn al-Mundhir said: the tasleem [in sujud as-sahw] is proven from more than one isnad (chain of narration), but there is some dispute concerning the tashahhud.” (Al-Mughni, 2/431-432)
According to Anl-Nawawi, among the things we learn from the hadith of Dhul-Yadayn are:
“That sujud as-Sahw was done at the time of the Prophet. That it consists of two prostrations, and that one should say takbeer (“Allahu akbar”) for each prostration. They are like the prostrations of prayer, because they are called sujud, and if they had been different, he would have explained that. One should say salam after doing sujud as-sahw, but there is no tashahhud, and if one has to do sujud as-sahw because of doing something extra in the prayer, this should be done after the salam.” (Sharh Muslim, 5/71).
Almighty Allah knows best.