The Messenger of Allah ﷺ would rest his head on the lap of one of us and recite the Quran while she was menstruating. One of us would also carry the prayer mat to the mosque and spread it out while she was menstruating.
أخبرنا محمد بن منصور عن سفيان عن منبوذ
كان رسول الله ﷺ يضع رأسه في حجر إحدانا فيتلو القرآن وهي حائض وتقوم إحدانا بالخمرة إلى المسجد فتبسطها وهي حائض
[Its chain of transmission is daif] Ahmad reported it (6/331) from Sufyan bin Uyayna with this wording, and he stated explicitly that he had heard it directly. The hadith also appears in Kubra, no. 267. As for Umm Manbudh, I did not find anyone who confirmed her reliability.