Praise be to Allaah.
There is nothing wrong with reciting Qur'aan in a state of minor impurity (without wudoo'), and some of the scholars have stated that there is consensus on its permissibility. There are saheeh hadeeths concerning this matter which demonstrate that it is not necessary to do wudoo' if one wants to recite Qur'aan and is in a state of minor impurity. Rather the difference of scholarly opinion has to do with touching the Qur'aan and whether one who is in a state of major impurity (janaabah) can recite it; the majority of scholars are of the view that both of these matters are not allowed.
It was narrated from Abd-Allaah ibn Abbaas (may Allah be pleased with him) that that he stayed one night with Maymoonah, the Mother of the Believers, who was his maternal aunt. I lay down across the width of the mattress and the Messenger of Allaah (blessings and peace of Allah be upon him) and his wife lay along its length. The Messenger of Allaah (blessings and peace of Allah be upon him) slept until halfway through the night, or just before or after that. The Messenger of Allaah (blessings and peace of Allah be upon him) woke up and started wiping the sleep from his face with his hand. Then he recited the last ten verses of Soorat Aal Imraan, then he got up and went to a waterskin that was hanging there, and did wudoo' from it and did it well, then he stood and prayed.
Narrated by al-Bukhaari, 4295; Muslim, 763
Imam al-Bukhaari included this hadeeth in a chapter entitled: Chapter on Reciting Qur'aan after Minor Impurity and otherwise.
Ibn Abd al-Barr (may Allah have mercy on him) said:
This indicates that Qur'aan may be recited without wudoo', because he slept for a long time, so there is no question that he woke up without wudoo'; then he woke up and recited Qur'aan before he did wudoo', then he did wudoo' after that and prayed.
End quote from al-Tamheed lima fi al-Muwatta' min al-Maaani wal Asaaneed, 13/207
Al-Nawawi (may Allah have mercy on him) said:
This indicates that it is permissible for one who is in a state of minor impurity to recite Qur'aan; this is the consensus of the Muslims.
End quote from Sharh Muslim, 6/46
And Allah knows best.