This source on the "Xiongnu Empire" seems to suggest a Turkic origin of the Xiongnu. I believe this is the most widely accepted theory among modern academics. The extent of mixture between Mongolic and Turkic peoples in the 3rd/4th centuries BC is debatable, but certainly much less than it was at the zenith of the Mongols in the 13th century AD.
The Xiongnu (Hongnu in Old Chinese, Xwn in Soghdian, probably Old Turkic Qun), also known as the Asiatic Huns, were one of the nomadic peoples of Ancient Central Asia. They're thought to have descended from various Turkic peoples known as Xianyun, Xunyu and Hongyu, yet all the knowledge we have come from Chinese sources written centuries later. However, as time passed, the name Xiongnu was applied to the Xiongnu’s subjects too, including Turkics, Mongolics, Tokharians, Iranics, etc.
The connection with the later European Huns is also a very tempting one, given the gap of a few centuries from their disappearance in Ancient Chinese's sources and appearance in late Roman histories.
The remnants of both Xiongnu empires lived as scattered throughout Western Turkestan for a long time, until they began migrating westwards around 350 AD. Under the leadership of their leader, Balamïr, they entered the territories of the Ostrogothic Kingdom in Ukraine in 375, and they founded the European Hunnic Empire (there are some scholars who doubt that the European Huns descended from the Xiongnu). Even though the majority of Xiongnu went to Western Turkestan, some Xiongnu stayed in Northern China where they set up small kingdoms after the fall of the Han Dynasty (Second Zhao, Xia, Northern Liang and Loulan were the Xiongnu kingdoms in Northern China).
The significant amount of detail provided by this source may suggest accurate information, but I personally would like to see citations, it being a secondary source. Nonetheless, several of the views reflected here are fairly widely accepted by modern historians (although with notable contention) from what I have read.