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