Surely people in nations like England, France, and the various kingdoms that made up what is now Germany, Italy and Spain would have heard of this terrifying menace from beyond the known world, sweeping across the Near East and then proceeding to rape and pillage Eastern Europe.
Did the nations of Western and Central Europe feel like they were a very serious threat? Was there even perhaps contact established with the Mongol hordes? What was the general consensus among the European nations that were less at risk than the likes of Russia, Poland and Hungary of the Mongols?
At first they thought they might be a good ally against the Muslim states in the Middle East in Egypt - this was the era of the crusades and supposedly the Mongols were Christian and would be glad to help. That turned out to be not quite accurate...and when the Mongols actually showed up in Europe they were terrifying and destructive.
This is a pretty vast topic so There is always much more that can be said, but I answered a similar question a couple of years ago: