Certainly, one is allowed to perform `Umrah during the months of Hajj, and there is nothing wrong with that at all. The Prophet (peace and blessings be upon him) is reported to have performed `Umrah four times, all of which were in the month of Dhul-Qi`dah.
Sheikh M. S. Al-Munajjid, a prominent Saudi Muslim lecturer and author, states the following: It is permissible to perform `Umrah during the months of Hajj, and there is no difference of opinion among the scholars concerning that. And it makes no difference whether you intend to perform Hajj in the same year or not.
The Prophet (peace and blessings be upon him) performed `Umrah four times, all of which were in the month of Dhul-Qi`dah, which is one of the months of Hajj, which are Shawwal, Dhul-Qi`dah and Dhul-Hijjah. And he did not perform Hajj except along with his last `Umrah, in the Farewell Pilgrimage.
Al-Bukhari (4148) and Muslim (1253) narrated from Anas (may Allah be pleased with him) that the Messenger of Allah (peace and blessings be upon him) performed `Umrah four times, all of them in Dhul-Qi`dah except for the one which he did with his Hajj. His `Umrah from Al-Hudaybiyah or at the time of Al-Hudaybiyah was in Dhul-Qi`dah, his `Umrah in the following year was in Dhul-Qi`dah, his `Umrah from Ji`ranah when he divided the spoils of Hunayn was in Dhu`l-Qi`dah, and his `Umrah which he did with his Hajj (was also in Dhul-Qi`dah).
An-Nawawi (may Allah have mercy on him) said in his commentary:
The conclusion we draw from the reports of Anas and Ibn `Umar is that they both indicate that the Prophet (peace and blessings be upon him) did four `Umrahs. The first was in Dhul-Qi`dah in the year of Al-Hudaybiyah, 6 AH, when they were prevented from going further so they exited ihram, but this was counted as a `Umrah for them. The second was in Dhul-Qi`dah in 7 AH, which was to make up for the previous year. The third was in Dhul-Qi`dah of 8 AH, which was the year of the Conquest of Makkah. And the fourth was with his Hajj, when he entered ihram for it in Dhul-Qi`dah but did the actions of `Umrah in Dhul-Hijjah.
An-Nawawi further said: The scholars said that the Prophet (peace and blessings be upon him) did `Umrah in Dhul-Qi`dah because of the virtue of this month and to go against the ideas of the jahiliyyah (pre-Islamic ignorance) because they used to think that this was one of the worst of evil actions. So he (peace and blessings be upon him) did it many times in these months to make it explicitly clear that this was permissible and to demonstrate clearly that the beliefs of the jahiliyyah were false.