Performing Umrah in Dhul Hijjah is permissible according to all scholars, as there is no prohibition against Umrah in any month of the year. The first ten days of Dhul Hijjah are among the best days of the year, and performing Umrah in them is virtuous. The question that generates more discussion is performing Umrah after completing Hajj, during the remaining days of Dhul Hijjah.
Sheikh Ibn Baz stated that performing Umrah after Hajj is permissible, as demonstrated by the Prophet's instruction to Abdur-Rahman to take his sister Aisha to Tan'eem for Umrah after their Hajj was complete. This occurred during the Farewell Pilgrimage in Dhul Hijjah. He considered this a clear evidence that Umrah in Dhul Hijjah — even after Hajj — is valid and carries reward.
Sheikh Al-Albani, however, noted that Aisha's Umrah was a special case — she was distressed because she had not been able to perform a separate Umrah (having been instructed to combine Hajj and Umrah as qiran). The Prophet accommodated her specific situation. He pointed out that the Prophet himself and the rest of the companions did not perform Umrah after Hajj. Based on this, Al-Albani considered it better to follow the Prophet's own practice of not performing a separate Umrah after Hajj, though he did not prohibit it.