Abu Ayyoub reported that the Messenger of Allah, salla Allahu alaihi wa sallam said “Whoever fasts the month of Ramadhan and then follows it by fasting six days during the month of Shawwal will be rewarded as if he had fasted the entire year. [Muslim, at-Tirmithi, Ibn Majah, Abu Dawood and Ahmad by way of Jabir].
A: Scholars differ regarding this issue; however, the soundest opinion is that it is prescribed to give precedence to observing Sawm of the missed days of Ramadan over the six days of Shawwal and other supererogatory Sawm, because the Prophet (peace be upon him) said: Anyone who fasts Ramadan and then follows it with six days of Shawwal it is as if they fasted the whole year. (Related by Muslim in his Sahih [Authentic Hadith Book]) If one chooses to observe Sawm of the six days of Shawwal first, it will not be considered as one who fasts these six days after fasting all of Ramadan, but after some days of Ramadan. Making up for the missed days of Ramadan is a Fard (obligatory act of worship) while fasting the six days of Shawwal is voluntary, and a Fard is more important to fulfill. May Allah grant us success!
Source: Islamic Research Journal
A: You have to hasten to fulfill the Kaffarah; it is not permissible to fast the six days of Shawwal before the Kaffarah, for fasting these six days is a Nafl (supererogatory act of worship) while the Kaffarah is a Fard (obligatory act of worship) which should be carried out immediately. It should be given priority over the Sawm of the six days of Shawwal or any other supererogatory Sawm.