Malik said, "No-one should carry the Qur'an by its strap, or on a cushion, unless he is pure. If it were permissible to do so, it would also have been permissible to carry it in its cover. This is not because there is something on the hands of the one who carries it by which the Qur'an will be soiled, but because it is disapproved of for someone to carry the Qur'an without being pure out of respect for the Qur'an, and in order to honour it." Malik said, "The best thing that I have heard about this is the ayat 'None touch it except the purified.' (Sura 56 ayat 79) It ranks with the ayat in Surat Abasa (Sura 80), where Allah, the Blessed and Exalted, says, 'No, it is a reminder, and whoever wishes will remember it. Upon honoured pages, exalted and purified, by the hands of scribes, noble and obedient.' "
USC-MSA web (English) reference: Book 15, Hadith 1