Performing Umrah in Dzulhijjah is boleh according to all scholars, as there is no prohibition against Umrah in any month of the year. The first ten days of Dzulhijjah 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 Haji, during the remaining days of Dzulhijjah.
Sheikh Ibn Baz stated that performing Umrah after Haji is boleh, as demonstrated by Nabi's instruction to Abdur-Rahman to take his sister Aisha to Tan'eem for Umrah after their Haji was complete. This occurred during the Farewell Pilgrimage in Dzulhijjah. He considered this a clear evidence that Umrah in Dzulhijjah — even after Haji — 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 Haji and Umrah as qiran). Nabi accommodated her specific situation. He pointed out that Nabi himself and the rest of the companions did not perform Umrah after Haji. Based on this, Al-Albani considered it better to follow Nabi's own practice of not performing a separate Umrah after Haji, though he did not prohibit it.