I would recommend checking out John of Plano Carpini's writings of his travels to the court of Ogedei Khan. https://depts.washington.edu/silkroad/texts/carpini.html

When he travelled to the Guyuk Khan's court he bore with him a letter from Pope Innocent IV called Cum Non Solum. In it, the Pope inquires as to the Mongols future intentions, requests that Christians be treated with respectfully and proposes peace between the Mongol Empire and Christendom. Of course, this led to some interesting inferences on the part of both parties as the letter, written in Latin, had to be translated something like 6 times to get into the Mongol language. Perhaps the most interesting miscommunication in the Pope's letter is that the word for "peace" in Mongolian is synonymous with subjection. The predominant belief in the Mongol ruling class was that all lands under the sky were subject to Mongol rule, even those lands that hadn't been conquered yet. Guyuk's response to Pope Innocent reads as follows:

"You must say with a sincere heart: "We will be your subjects; we will give you our strength". You must in person come with your kings, all together, without exception, to render us service and pay us homage. Only then will we acknowledge your submission. And if you do not follow the order of God, and go against our orders, we will know you as our enemy"

That said, the predominant attitude in the Mongol empire toward religion was, to borrow a not-so-precise modern term, tolerance. That is to say, the Mongolians believed in taking out what could be called "celestial insurance" in that they didn't really care who their subjects worshipped so long as they prayed for the victory and wellbeing of the Great Khan and the empire.