Sheikh `Umar al-Muqbil, professor at al-Imâm University
There is no authentic hadîth in this regard. There are a few hadîth that indicate it is disliked to lie down on one’s stomach. However, these hadîth are all weak.
The best among these hadîth is the one related by Ahmad from al-Shurayd b. Suwayd from Ibrâhîm b. Maysarah that`Amr b. al-Shurayd heard the Prophet (peace be upon him) say that it is forbidden to lie on one’s stomach.
The narrators of this hadîth are all truthful, but this hadîth does not actually reach all the way back to the Prophet (peace be upon him) with an unbroken chain of transmission. The reason for this is that `Amr b. al-Shurayd was one of the Successors who met the Companions but never met the Prophet (peace be upon him). [al-Taqrîb (5049)]
Therefore, we cannot say that this manner of lying down is forbidden or even disliked in Islam.
However, the customs of the people are supposed to be considered in such matters, and this manner of lying down could very well be something rude or offensive in some societies. In such cases, it would be better to avoid it.
And Allah knows best.
Source: Islam Today