This is a well-known area of legitimate ikhtilaf. The Shafi'i, Maliki and Hanbali schools hold that it breaks wudu, relying on the hadith of Busrah bint Safwan that the Prophet (ﷺ) said, "He who touches his penis should perform ablution" (Sunan Abi Dawud 181, sahih; Sunan al-Tirmidhi 82, which adds "...he is not to pray until he performs wudu"). The Hanafis and others hold that it does not, relying on the hadith of Talq ibn 'Ali, in which a bedouin asked the Prophet (ﷺ) about a man who touches his penis after performing wudu and he replied, "That is only a part of you" (Sunan Abi Dawud 182; Sunan al-Nasa'i 165). Many scholars reconcile the two by saying that a direct, skin-to-skin touch — especially with desire — breaks wudu, while an incidental touch through clothing does not. The same discussion applies to a woman touching her own private part. Since both hadiths are authentic and both positions are held by major imams, the prayer of someone following either view is valid; the more cautious course is to renew wudu after a bare-handed touch.
Q&A · Purification
Does touching your own private parts break wudu?
Evidence — the daleel
On deen2u
Informational, not a personal fatwa. Consult a qualified scholar for rulings on your situation.