Warning: historians understand these letters to be a literary creation, produced well after the actual conquest period. That said, they can help us talk about the ways in which later generations understood the conquests.
A 9th-century Description
This passage is from the history of Yaʿqubi, who wrote in the 9th century.
Source, pp. 685-7.
A 8th/9th-century Description
This passage relates the purported conversation between the Byzantine emperor Heraclius and Abu Sufyan, the leader of the Umayyads. It begins with the emperor summarizing their conversation about the Prophet Muhammad. This account comes from the book by ‘Abd al-Razzaq b. Hammam al-Himyari, who lived in the 9th century, but it preserves the reports of Ma’mar b. Rashid, who lived in the 8th.
Source, pp. 47-9
The Purported Letters
Over the years, people have claimed to find the actual letters that Prophet Muhammad sent to the heads of state. These are not authentic, but they show the lasting interest in the story. This image is a 20th-century drawing of the purported letter from the Prophet Muhammad to the governor of Egypt.
Medieval Islamic History 