This issue is known to the scholars as observing naafil fasts before making up missed Ramadaan fasts. There is a difference of scholarly opinion concerning this matter. Some scholars say that it is haraam to observe naafil fasts before making up days that one owes, because it is more important to start with an obligatory action than a naafil one. Some of the scholars say that it is permissible.
Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have mercy on him0 was asked about combining making up a missed obligation and doing something that is mustahabb: is it permissible for a person to do the mustahabb action and make up the obligatory action later on, or should he do the obligatory action first, such as fasting the day ‘Ashoora’ which coincides with making up a missed Ramadaan fast?
With regard to obligatory and voluntary fasts, what is prescribed in sharee’ah and what makes sense is to start with the obligatory fasts and then do the naafil ones, because the obligatory fast is a debt which must be paid, whereas the naafil fast is voluntary and is to be done if one can manage it, otherwise there is no sin on one. Based on this, we say to the one who owes a missed Ramadaan fast: make up what you owe before you observe a voluntary fast. If he observes a voluntary fast before making up what he owes then the correct view is that his voluntary fast is valid so long as there is still enough time to make up the missed fasts, because a person may make up missed Ramadaan fasts so long as there is still enough time for him to do so before the next Ramadaan comes. So long as there is still plenty of time, it is permissible for him to observe voluntary fasts. This is similar to the case of obligatory prayers, such as if a person offers a naafil prayer before an obligatory prayer when there is still plenty of time, this is permissible. Whoever fasts on the day of ‘Arafah or ‘Ashoora’, and still owes some days from Ramadaan, his fast is valid. But if he has the intention of fasting this day to make up for a missed Ramadaan fast, he will have two rewards – one for the day of ‘Arafah or ‘Ashoora’ and another for making up the missed fast. This applies to all voluntary fasts that are not connected to Ramadaan. With regard to fasting the six days of Shawwaal, these are connected to Ramadaan and can only be done after making up missed Ramadaan fasts. If a person fasts these days before making up his missed Ramadaan fasts, he will not get the reward for them, because the Prophet (peace and blessings of Allaah be upon him) said: “Whoever fasts Ramadaan then follows it with six days of Shawwaal, it will be as if he fasted for a lifetime.” It is known that the one who still has some days to make up is not regarded as having fasted Ramadaan until he makes up those days. Some people think that if they fear Shawwaal is going to end soon and they fast the six days even though they still owe some days from Ramadaan, that this is acceptable. This is a mistake, because these six days cannot be fasted until a person has made up the days that he owes from Ramadaan.
Majmoo’ Fataawa Ibn ‘Uthaymeen, 20/438.
Based on this, it is permissible for you to fast the first ten days of Dhu’l-Hijjah as a naafil fast, but it is better for you to fast them with the intention of making up what you owe of Ramadaan, and you will have two rewards in sha Allaah.
See also question no. 23429.
And Allaah knows best.