The time for a Sunnah prayer that comes before a fard prayer does not begin until the time for that prayer begins.
Ibn Qudaamah said in al-Mughni (2/544)
The time for every Sunnah prayer that comes before a fard prayer lasts from when the time for that fard prayer begins until it is done. The time for every Sunnah prayer that comes after a fard prayer lasts from after the fard prayer is done until the time for that prayer ends. End quote.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him) was asked:
I entered the mosque for Fajr prayer and prayed two rak’ahs, and when I stood up for the second rak’ah, the muezzin began to give the adhaan for prayer. I had intended that prayer to be the Sunnah of Fajr, since when I left my house the adhaan was being given in some of the mosques. When I had finished that prayer, I sat and read Qur’aan, and someone next to me told me to get up and pray the Sunnah of Fajr. I told him that I had already prayed, and he said, “That is not permissible, you have to pray it again, because the muezzin gave the adhaan whilst you were praying.” I hope that you can advise me about that.
If the muezzin who gave the adhaan whilst you were praying had delayed the adhaan and your prayer coincided with it after dawn had broken, then you have performed the Sunnah and that is sufficient and there is no need to repeat it. But if you are uncertain about that and you do not know whether the muezzin who gave the adhaan whilst you were praying did so after dawn had broken or when it was breaking, then it is more on the safe side and better for you to repeat the two rak’ahs, so that you will have done them after dawn for certain. End quote.
Majmoo’ Fataawa al-Shaykh Ibn Baaz, 11/369, 370
Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have mercy on him) was asked:
Is it essential with regard to the regular Sunnah prayers that are offered before Zuhr and Fajr that the time for the (fard) prayer should have begun?
The regular sunnah prayers that are offered before a fard prayer must be done after the time for the fard prayer begins. If we assume that a person offers them before the time begins, thinking that the time has begun, then he finds out that the time had not begun, then he must repeat them, and the first prayer will be a general kind of naafil prayer, not a regular Sunnah prayer. End quote.
Liqaa’aat al-Baab il-Maftooh, question no. 590
And Allaah knows best.