Answer/Question/looking for confirmation here, hope this is OK. I read Genghis Khan and the Making of the Modern World by Jack Weatherford, who on the one hand seems to have decent credentials, but the book, and all his books, have a very pop history vibe to them.

One of the most interesting things in the book (if true) was the way some Europeans equated the Tatars - which were a tribe that was absorbed into the Mongols early on and made up a sizable chunk of their force - with "Tartarus," a Greek version of Hell. European sources would often refer to the Mongols as a whole as Tartars (not Tatars, notice the extra "r") and imply that they were sent from this mythical hell to punish Europeans for their sins. The Mongols tended to be pretty brutal to those who did not submit, so their reputation certainly would have preceded them.

I know you asked about the 13th century, but beyond the initial conquest, he argues that given this vast empire which connected east and west, so much wealth flowed into Europe from Asia that Genghis Khan became this legendary king with fabled wealth and power even in the European mind. He is mentioned in Canterbury Tales as such iirc and there were various other examples that I've forgotten.

Not an expansive answer, but that's what I have. As I said before, I was kind of hoping for someone to add on to this and confirm what I read since I was always a little suspicious of this particular book.

Ok here we go, this is all from "The Mongols, (The Peoples of Europe)" by David Morgan, a book I can thoroughly recommend if you want to learn about the Mongols and especially their relationship with Europe as can be guessed by the title.

Initially "No one had heard of the Mongols [in Europe]. Chingiz Khan's conquests in central asia and china, unsurprisingly, went unnoticed."

"The first European to have brought back accurate information about the Mongols was friar Julian of Hungary, who traveled east as an emissary of king Bela IV of Hungary in 1234-5 and 1237. On the latter occasion his mission was interrupted by the beginnings of Batu's Russian Champaign." (The Mongols invaded the principalities of Rus starting in 1237, Batu Khan was the representative of the Mongol royal family that lead the invasion, though it was Subotai the general who actually conducted the invasion.)

This is what you are looking for I think (most of this comes from a contemporary monk and chronicler Mathew Paris, quoted in "The Mongols" by David Morgan):

"The invasion of eastern Europe woke Christendom up very sharply, as we can see from the quantity of letters from a number of concerned clergy and princes...These include a circular letter from Frederick II (The Emperor of the Holy Roman Empire) to his fellow monarchs, urging them to assemble forces and to unite against this new peril"

The Europeans also came up with a bunch of different explanations as to how and why the Mongols had appeared as chronicled by Mathew Paris: "They were the lost ten tribes of isreal, albeit lacking any knowledge of Hebrew; or it was a plot hatched by the emperor Frederick (HRE emperor from before), his object being to obtain the homage of the kingdom of Hungary. The last allegation chocked the chronicler: Gof forbid' he exclaimed, 'that so much wickedness should be lurking in any one mortal body"

"It fell to innocent IV, after his election to the papacy in 1243" (one year after the end of the Mongol invasion of Poland and Hungary) " to take effective action to clarify the situation. The question of the Mongols was placed on the agenda of the Council of Lyons in 1245. and the result was the dispatch of three embassies to Mongol territory. ...The Pope's pleas for moderation in Mongol conduct fell on deaf ears, and the message that came back to Europe was an uncompromising one: the pope, with the other European potentates, was required to present himself forthwith at the court of the great khan to make his submission,"

Though the relationships between the Mongols and the Europeans were not all negative, and in fact when the Mongols first invaded Muslim Persia many Christians thought that the Mongols were going to help them reclaim the holy land.

The thought of the Mongols helping eurpeans conquer toads the holy land did not go away and in 1248 King Louis IX of France saw some Mongol emissaries in the hope of forging an alliance to assit him in his crusade to Egypt. In reality the Mongols were likely never going to actually help the Christian soldiers (The emissaries lied to the French and told them that Guyuk Khan had converted to Christianity) and a year later the Mongol political system changed, and the new Khatun was not willing to entertain the King of France's hopes. Instead of anything like the possible alliance that had been discussed, the Khatun actually demanded King Louis's submission, and Louis saw the diplomatic mission as a waste of time and money. "His friend and biographer Joinville tells us that 'the King, you may be sure, repented of having sent him (The great Khan) his ambassadors'".

Finally, after the Mongol Empire broke apart the IL khanate considered a French alliance to assist them in fighting either the Mameluke Sultanate or the Golden Horde, both of which were Muslim.

Sorry about the fairly poor writing, I hope you found the information interesting. There were some spots I was really brief on so if you have any questions about any particular part or anything else about the Mongol Empire please ask.