Yes, there are many historical records and strong evidence on both sides that verify this. But they were not just aware of each other, they in fact sought to actively trade with one another, which is how the Silk Road came about. The vast distance between the Romans and the Chinese meant direct trade was impossible, so the Silk Road came about as an indirect, looping trade network.

So when were the Roman Empire and Ancient China aware of each other? Evidence shows that first contact can be traced back to the Han Dynasty (206 BC–220 AD). Interestingly, while both sides knew the other lay across a vast distance, direct contact was either never established or has yet to be proven. The distance between the two was a large stretch of land consisting of basically the entirety of India and large parts of the Middle East. The Indian kingdoms, and the Parthian Empire that was regnant in the Middle East at the time, are said to have purposely inhibited contact between China and Rome so they could maintain a monopoly on the lucrative Silk Road trade between the two (direct trade between the two would remove the need for an intermediary like the Parthians).

The Chinese, at least, did try to establish direct contact though. Gan Ying was a Han envoy sent on a mission to Rome, but the furthest west that he made it was either the Persian Gulf, the Black Sea, or perhaps the Mediterranean's southwestern coastline (this is unclear; sources only translate into 'the western sea', meaning it could be any of the three, but it is unlikely to be the Mediterranean due to Rome's influence over the region meaning Gan Ying could have easily gone to Rome if he had reached the Mediterranean coastline). It is theorized that the Parthians dissuaded Gan Ying from going any further west, citing dangers of travel + a lengthy return journey if the winds were unfavorable, but it is possible they did so upon learning of his purpose and wanted to restrict contact between Rome and China.

Still, Gan Ying's failed journey to Rome didn't amount to nothing. He did gain valuable information about Rome, and knew, more or less, where Rome was, as his travels had brought him past the northern fringes of India into the Parthian Empire, where there was active trade with the Romans. He learned what he could in the Parthian Empire, and then left back home to China, where his findings were recorded in the Hou Han Shu (後漢書), or the Book of the Later Han. An important piece of evidence pointing to Chinese awareness of Rome (which is listed as "Da Qin", or Great Qin) is recorded within, as follows (translated):

(Da Qin)'s territory extends for several thousands of li. It has more than four hundred walled towns. There are several tens of smaller dependent kingdoms. The walls of the towns are made of stone. They have established postal relays at intervals, which are all plastered and whitewashed.

Hope this helped.

Source: Fan, Ye (September 2003). Hill, John E., ed. "The Western Regions according to the Hou Hanshu: The Xiyu juan, "Chapter on the Western Regions", from Hou Hanshu 88, Second Edition (Extensively revised with additional notes and appendices): Section 11 – The Kingdom of Daqin 大秦 (the Roman Empire)". Depts.washington.edu. Translated by John E. Hill.