Europeans were well aware of the Muslims before they reached Europe, and their reactions ranged from initial excitement that the Mongols were a Christian army coming to help them crush Islam, to their sheer terror when the Mongols actually showed up. The arrival of the Mongols was probably the most terrifying disaster they had experienced until the Black Death a hundred years later (which may have also been indirectly caused by the Mongols!).

Europe had heard rumours about the Mongols early in the 13th century. There were still crusader states on the eastern coast of the Mediterranean coast, and there was a crusade against Egypt in 1217-1221, so the crusaders heard news about the Mongols and reported it back in Europe. Jacques de Vitry, the bishop of Acre int he crusader states and a participant in the crusade against Egypt,

“…expressed hope that ‘the numerous Christian monarchs who dwell in the East as far as the territory of Prester John’ would aid the crusaders by attacking the Muslims. Consequently, when in 1221 the crusading leaders heard fresh rumours of operations against the eastern Muslims by an allegedly Christian army, they too were easily persuaded to link them with Prester John.” (Jackson, The Mongols and the West, pg. 21)

“Prester John” was a legendary eastern Christian king from…somewhere. Ethiopia? India? Somewhere further east? They really had no idea. This was also sometimes conflated with the legend of an eastern “King David”. In 1221 they were probably hearing rumours about Genghis Khan himself. At the same time as the crusade, Genghis destroyed the Khwarizmian Turks in Central Asia, and many of them migrated west towards Syria, so the crusaders (and then Europe in general) knew there was *someone* out there attacking Muslims.

Nothing arrived immediately though, and Genghis died in 1227. When the Mongols actually arrived in Europe, they took a different route, from the north through Russia. The Mongols of the Golden Horde conquered Novgorod and Kiev and the other Russian principalities in 1237. The Russians remained subjects of the Golden Horde for another two centuries. The Mongols then sent messengers ahead to Poland, Hungary, and the Holy Roman Empire, demanding that they all submit to the great khan. The Hungarians, who were also originally central Asian nomads, understood what was about to happen. Some fellow Asian nomads, the Cumans, fled from the Mongols, through Russia, and settled in Hungary, which was extremely disruptive for Hungarian society. But a bit further west, the Holy Roman Emperor (Frederick II) didn’t think they were much of a threat, and didn't really prepare for them at all.

The Mongols reached Poland and Hungary in 1241, and even as far as Croatia and Austria in 1242, before mysteriously retreating. The traditional explanation is that Ogodei Khan (Genghis’ son) died and had they had to go back home to elect a new Khan, but it could have been due to the weather in Europe, or disease, or some other reason. In any case, Hungary was completely devastated and there was no reason the Mongols couldn’t have devastated the rest of Europe as well, if they had kept going. European rulers tried to raise money and armies to fight the Mongols, but then they left and the threat seemingly disappeared. They did come back eventually - they raided Poland and Hungary again in the 1260s, but they weren’t as massive a threat as they were in the 1240s.

The Mongols also invaded the Seljuk Sultanate of Rum in Anatolia, and subjugated Georgia and Armenia in the Caucasus. In Anatolia they encountered crusaders and Byzantines - the Byzantine state in Nicaea was trying to take back Constantinople, which was controlled by the crusader Latin Empire at the time. Neither of them really fought against the Mongols, but both allied with them against the Seljuks.

The crusader states in the Near East also eventually encountered the Mongols in the 1250s. The Khwarizmians displaced by the Mongols in 1221 slowly moved west and sacked Jerusalem in 1244. They allied with the Ayyubid Muslims in Egypt and destroyed the crusader army at the Battle of Forbie later in 1244. This led to another crusade against Egypt led by Louis IX of France. Louis’ crusade was also destroyed and he was taken prisoner - and while he was in prison in Cairo, the Ayyubids were overthrown by their Mamluk slave-soldiers, who established their own dynasty in Egypt and Syria.

The Mongols slowly followed the Khwarizmians westward and Hulegu Khan, one of Genghis’ grandsons, destroyed Baghdad in 1258. The crusaders hoped Hulegu would join them in fighting the Mamluks, but he was completely uninterested, aside from forcing them to submit to Mongol authority. The crusader principality of Antioch was subjected to Hulegu, at least temporarily. In 1260, the Mongols captured Aleppo and Damascus as well, but later that year, the Egyptian Mamluks defeated them at the Battle of Ain Jalut, which mostly ended the Mongol presence in Syria. They still ruled the Ilkhanate in Persia though.

Meanwhile, in Europe, the pope and other authorities decided they should send ambassadors to the Mongols to negotiate with them, and to try to convert them to Christianity as well. Some of the Mongols were actually Christians already, and European Christians were well aware of that, although Latin Christianity and eastern (sometimes called “Nestorian”) Christianity were a bit different.

In 1245, Pope Innocent IV sent some ambassadors to the Mongol Khan, Guyuk, at Karakorum. They explained the basics of (Latin) Christianity and encouraged him to convert. Guyuk responded:

"Thou thyself, at the head of all the Princes, come at once to serve and wait upon us! At that time I shall recognize your submission." (Allen and Amt, pg. 369)

A few years later, Louis IX also sent ambassadors while he was on crusade, but again, the Mongols were not interested unless he was willing to submit to them first. One of the missionaries was William of Rubruck, who, among other things, tried to explain Latin Catholicism to the Nestorian Christians, but they just laughed at him. Why should they submit to Rome when the Mongols were already the masters of the world? Clearly their own version of Christianity was superior!

Commercial contact went a bit better. By this point in the 13th century, there were already westerners living in Asia (thanks to the crusader states) and some had travelled as far as China and the Mongol territories. By the 1280s there were lots of Italian merchants in the Ilkhanate, and there were now also Genoese colonies in the Black Sea that had contact with the Golden Horde in Russia (which is where thee Black Death came from in the 14th century). This was all before Marco Polo - the Polos were able to travel that far east because so many other Europeans had already been there and the trade routes were already well known.

So, yes, Europeans were certainly well aware of the Mongols, and there was quite a lot of contact between them. Europeans tried to negotiate with them, and convert them, but the Mongols were not interested at all. Europeans were absolutely terrified when the Mongols suddenly showed up, devastated Eastern Europe, and just as suddenly disappeared.

Sources:

Peter Jackson, The Mongols and the West (Routledge, 2005)

David Morgan, The Mongols, 2nd ed. (Blackwell, 2007)

Peter Jackson, The Mission of Friar William of Rubruck: His Journey to the Court of the Great Khan Möngke, 1253-1255 (Hakluyt Society, 1990)

S.J. Allen, Emilie Amt, eds., The Crusades: A Reader, 2nd ed. (University of Toronto Press, 2014)