Verbatim · Ayatollah A. Sistani
There is no problem in the recitation or memorisation of the holy Quran in such situations, except for the verses for which prostration (Sajdah) is obligatory. These verses are found in four chapters of the Quran: Surat al-Sajdah (Chapter 32) Verse 15 Surah Fussilat (Chapter 41) Verse 37 Surah al-Najm (Chapter 53) Verse 62 Surah al-Alaq (Chapter 96) Verse 19
Ayatollah A. Sistani sistani.org Q&A #26337 other marjas may differ
Note the four verses of obligatory sajdah specifically. Surat al-Sajdah 15, Surat Fussilat 37, Surat al-Najm 62, Surat al-Alaq 19. Touching the Arabic text of the Quran with any part of the body is separately forbidden during hayd; recitation from memory or sight is what’s discussed here.