Can You Pray Sunnah before Adhan?
Question

Should the Sunnah of Fajr be offered after the time for Fajr has begun, or should it be done after the Adhan – which is given before the time for the prayer begins?
Praise be to Allah.The time for a Sunnah prayer that comes before an obligatory prayer does not begin until the time for that prayer begins. 
Ibn Qudamah (may Allah have mercy on him) said in Al-Mughni (2/544) 
“The time for every Sunnah prayer that comes before an obligatory prayer lasts from when the time for that obligatory prayer begins until it is done. The time for every Sunnah prayer that comes after an obligatory prayer lasts from after the obligatory prayer is done until the time for that prayer ends.”
Shaykh `Abd Al-`Aziz ibn Baz (may Allah 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 Mu’atdhdhin began to give the Adhan for prayer. I had intended that prayer to be the Sunnah of Fajr , since when I left my house the Adhan was being given in some of the mosques. When I had finished that prayer, I sat and read Quran, 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 Mu’atdhdhin gave the Adhan whilst you were praying.” I hope that you can advise me about that. 
He replied: 
“If the Mu’atdhdhin who gave the Adhan whilst you were praying had delayed the Adhan 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 Mu’atdhdhin who gave the Adhan 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.” (Majmu` Fatawa Ash-Shaykh Ibn Baaz, 11/369, 370) 
Shaykh Muhammad ibn Salih Al-`Uthaymin (may Allah have mercy on him) was asked: 
Is it essential with regard to the regular Sunnah prayers that are offered before Thuhr and Fajr that the time for the obligatory prayer should have begun? 
He replied: 
“The regular sunnah prayers that are offered before an obligatory prayer must be done after the time for the obligatory 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 Nafil (supererogatory) prayer, not a regular Sunnah prayer.” (Liqa’at Al-Bab Al-Maftuh, question no. 590) 
And Allah knows best.