There are cave paintings by what are considered 'proto-mongol' peoples. There are seeming overlaps between previous steppe people, like the Xionghu, and Mongol culture. Plus, some scholars even hypothesize that there is an ancestral linkage there. Nomadic Mongols and their more settled Mongol co-dependents were in regular economic contact with the outside world, but this doesn't mean that the writings of say, a Han scholar would be available to them.

After conquering the Chinese, it just sort of seems likely that medieval Mongols would have at the very least become knowledgeable of the history of China's perpetual strained relations with steppe peoples, like the Xiongnu. Now, my example below should show you why not to trust intuition too much.

I have read within the work "The Horse-head Fiddle and the Cosmopolitan Reimagination of Mongolia" by Peter Marsh mention of a so called "Xiongnu Fiddle". As you may know the horse had fiddle is a integral part of Mongol folk music. So this "Xiongnu Fiddle" has been held up as a cultural or linguistic linkage between the two, but as Marsh explains: "His idea that the Xiongnu word kong is related with the Mongolian word khuyur through the phoneme ko-kho is a misunderstanding of Mongolian etymology and linguistic theory. Nor did Badraa make any attempt to show anything other than a similarity between two words in the language of two people separated by many centuries. Furthermore we may prsume that the words used in the Chinese historical sources were CHINESE inerpretations of Xiongnu speech, which raises further doubts about Badraa's interpretion. In the end his evidence and analysis are much to weak to prove any DIRECT connection to much less the existence of, a "Xiongnu Fiddle".

And he goes on to say that uncritical scholars took this mistake and ran with it: "In his history about the Xiongnu, the historian G Sukhbaatar reported Badraa's research "has made it clear the Xiongnu had a Fiddle"." This should give you some idea of how even scholars can get twisted up in connections based on the faintest glimmer of pseudo similarity. https://books.google.com/books?id=gjuTAgAAQBAJ&pg=PA126&lpg=PA126&dq=xiongnu+instruments&source=bl&ots=z43AwQibGk&sig=CxGpWyeH2E3YQbz2QPOw3fOQVt8&hl=en&sa=X&ved=0ahUKEwjHz8GU9JTWAhUG5SYKHTqIBnkQ6AEIOTAG#v=onepage&q&f=false

In the Encyclopedia of Mongolia and the Mongol Empire by Chris Atwood he strongly suggests that there is a technological relationship between the Xiongnu and the Mongol.

"The Mongol army inherited many centuries of military development. The warriors of the earliest nomadic empire in Mongolia, the Xiongnu, had only simple saddles, no stirrups, and a bow inferior to that of their Chinese enemies. By the sixth century Byzantine descriptions of the Avars describe the full panoply of the heavily armored Inner Asian cavalryman: wooden saddle and iron stirrup giving a strong seat on an armored horse for shots with a powerful composite bow, numerous remounts and flocks of sheep accompanying the warriors, and characteristic tactics involving ambushes, sudden appearance and disappearance, and feigned retreats followed by a volley of arrows and a sudden charge."

As far as oral tradition goes, one of the older Mongol story songs is preserved in Chinese, "Song of the Xianbei Brother." The Muren (Mongolic people) have song going back at least to around the time of the 1st century CE. This roughly coincides with the Xiongnu "Empire", so if we dare to be like Sukbataar and Badraa, we might dare to presume that the answer to your question is probably yes in the case of the Xiongnu.