Bismillaah:

‘Aishah رضي الله عنها‎‎ relates:

The Prophet صلي الله عليه وسلم would rest his head in my lap and recite Qur’aan.

Takhreej:

{ Imam Al-Bukhaari #7549 and Muslim #301 fe Saheehain }

Shaykh al Fawzaan حفظه الله commented:

There is nothing wrong with reading the Qur’aan when lying down, whether one is lying on a bed or on the ground. There is nothing wrong with that. A person may recite Qur’aan whatever position he is in, standing or sitting or reclining, and whether he has wudoo’ or is in a state of minor impurity, if he is reciting from memory. But if he is reading from the Mus-haf, then it is not permissible for the one who does not have wudoo’ to touch the Mus-haf unless he has done wudoo’.

{ Al Muntaqa min Fataawa }

As for Shaykh's statement " it is not permissible for the one who does not have wudoo’ to touch the Mus-haf unless he has done wudoo " this is a subject of great discussion and differences of opinion among the Scholars.

*** And Allaah عزوجل knows best ***