In Ghusl (bath for purification) whole body is washed with water and rinsed by our hands. In the process we touch our penis also. So will my Wudu remains valid for prayer or should I make new Wudu for it? Does touching the penis (private part) invalidate Wudu? Please specify clear steps for performing Wudu.

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

The majority of the Muslim scholars are of the opinion that touching the private parts directly by the palm of the hand and fingers; i.e., without having any partition between the palm and the private parts nullifies Wudu (ablution) for the Hadeeth of Busrah that the Prophet said: “Whoever touches his sexual organ should perform Wudu.” [Abu Daawood, Ibn Maajah and Maalik in Al-Muatta’]

Imam Al-Bukhaari said: “This is the most authentic Hadeeth in this subject.”
‘Awn Al-Ma'bood Sharh Sunan Abu Daawood reads: “The above Hadeeth establishes that touching the sexual organ annuls ablution…”
The same opinion is held by ‘Umar ibn Al-Khattaab, his son ‘Abdullah, Abu Ayyoob Al-Ansaari, Abu Hurayrah, ‘Abdullah ibn ‘Amr ibn Al-'Aas, Jaabir, ‘Aa’ishah, Umm Habeebah, Busrah and others, may Allah be pleased with them.
Therefore, whoever touches his sexual organ after completing Ghusl should repeat the ablution.

Allah knows best.