There is nothing wrong in reciting the Qur’an while performing Tawaf around the Ka`bah.
In this regard, we cite what Sheikh Sayyed Sabiq states in his well-known book, Fiqh As-Sunnah:
A person may recite the Qur’an while performing Tawaf, because the purpose of Tawaf is to remember Allah and the Qur’an is a reminder and a means of remembering Him. `A’ishah reported that the Prophet (peace and blessings be upon him) said: “Tawaf (circumambulation) around the House of Allah, walking between Safa and Marwah, and throwing the pebbles at the Jamarahs are instituted for the remembrance of Allah.” (Abu Dawud and At-Tirmidhi)