All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
There are five forbidden times for performing voluntary prayer as have been clarified in detail in the Hadeeth of ‘Uqbah ibn ‘Aamir as he said, "The Messenger of Allaah, sallallaahu ‘alayhi wa sallam, forbade us to pray or to bury our dead during three particular times: when the sun rises until it is high in the horizon, when it is at its meridian until its goes past it and the shadows of objects start to appear and when it starts to set until it sets completely .” [Muslim]
Moreover, Abu Sa‘eed Al-Khudri reported that the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, said: “There is no prayer after Fajr prayer until sunrise and there is no prayer after ‘Asr prayer until sunset.” [Al-Bukhari and Muslim]
These five times are the forbidden times of performing voluntary prayer. As for the voluntary prayers that are performed for a specific reason such as Salatul-Kusoof (i.e. eclipse prayer), and Tahiyyat-ul-Masjid (i.e. salutation to the mosque) , the preponderant opinion in this regard is that it is permissible to perform them at these times.
If you are unable to stay at the mosque after the Fajr prayer until after the sun has risen above the horizon by the length of a spear (i.e. about a quarter of an hour after sunrise) in order to perform the two-Rak‘ah Dhuha prayer so as to gain the rewards of performing ‘Umrah and Hajj in perfect completion, then it is impermissible for you to perform the Dhuha prayer before sunrise, since this is one of the forbidden times of prayer as mentioned above.
Allaah Knows best.