All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
Prophet Mohammad (PBUH) has urged Muslims to do good deeds on the first ten days of Dhul-Hijja. Ibn `Abbas narrated: The Prophet (PBUH) said, "No good deeds done on other days are superior to those done on these (first ten days of Dhul Hijja)." Then some companions of the Prophet (PBUH) said, "Not even Jihad?" He replied, "Not even Jihad, except that of a man who does it by putting himself and his property in danger (for Allah's sake) and does not return with any of those things."{Related by Bukhari}.
The aforementioned Hadith has generalized good deeds. Therefore, a Muslim may recite Quran, praise Allah, maintain the bonds of kinship, or observe fast. Al-Hafiz Ibn Hajar stated: "The preference of the first ten days of Dhul-Hijja arises from the fact that many great acts of worship coincide on them, such as prayer, fast, charity, and pilgrimage, which aren`t likely to coincide on other days."{Fateh Al-Baree, 3/390}.
The word "Days" {A day extends from break of dawn to sunset}-in the above Hadith- indicates that fasting is the best good deed to do in the day time while optional prayer is the best good deed to do at night, and this is why night prayer was legislated in the last ten days of Ramadan. Based on this, fasting the first ten days of Dhul-Hijja is recommended.
Moreover, it was reported that the Prophet (PBUH) used to fast the first ten days of Dhul-Hijja on constant basis. One of the wives of the Prophet (PBUH) narrated: Hunaydah ibn Khalid narrated from his wife on the authority of one of the wives of the Prophet (PBUH) who said: The Messenger of Allah (PBUH) used to fast the first nine days of Dhul-Hijjah, Ashura' and three days of every month, that is, the first Monday (of the month) and Thursday."{Related by Ahmad, An-Nassai`, and Ibn Hibban graded it as Sahih}. However, scholars of Hadith stated that none of the narrations pertaining to the virtue of fasting the first ten days of Dhul-Hijja is authentic (Sahih). Please refer to {Nasb al-Rayah, 2/156}. Nonetheless, this doesn`t prevent people from fasting those days since the former Hadith, related by Bukhari, has generalized the virtue of offering good deeds on them, such as fasting.
As regards the narration related by Muslim where Aisha (May Allah be pleased with her) said: "I never saw the Messenger of Allah (PBUH) fasting in the ten days of Dhu'I-Hijja," al-Imam An-Nawawi said: "Scholars said that this Hadith gives the false impression that it is disliked to fast the first ten days of Dhul-Hijja, which are actually the first nine days of Dhul-Hijja. They added that this is open for interpretation, so there is no reprehensibility in fating those nine days. Rather, it is very much recommended; particularly, the ninth day: the Day of Arafah, and the virtue of fasting it was mentioned in many a Hadith……What Aisha said could mean that the Prophet (PBUH) didn`t fast those days due to being sick, on a journey or else, or that she didn`t see him doing so although he has actually fasted them. This interpretation is supported by the Hadith of Hunaidah Ibn Khalid who narrated from his wife on the authority of one of the wives of the Prophet (PBUH) who said: The Messenger of Allah (PBUH) used to fast the first nine days of Dhul-Hijjah, Ashura' and three days of every month, that is, the first Monday (of the month) and Thursday."{Shareh An-Nawawi, 8/71}.
"It is probable that the Prophet (PBUH) didn`t fast those days for fear that Muslims would think that it is an obligation."{Fateh al-Barie, 2/460}.
In conclusion, it is recommended that a Muslim observe fast in the first ten days of Dhul-Hijja in line with his/her ability since there is a great reward for offering voluntary fast. Rather, combining fast with the virtue of the ten days of Dhul-Hijja doubles the Muslim`s reward from Allah, The Almighty. If we assume, for the sake of argument, that there is no authentic Hadith proving that the Prophet (PBUH) has fasted the first ten Days of Dhul-Hijja, this doesn`t mean that there is no virtue in fasting them. And Allah knows best.