During Tawaf, one is free to engage in Dhikr and Du’a. Reading the Quran is the perfect form of dhikr. Therefore, one may read it from memory after making the essential Duas associated with Tawaf.
However, one should avoid reading from the Mushaf in large crowds. By doing so, one may inconvenience others. It may also amount to dishonoring the Holy Book.
Allah Almighty knows best.
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