Buddhism was actually actively spreading westward and was well established in the region of modern day Afghanistan up until the 7th century when it was abruptly halted by the arrival of Islam which ended up dominating the religious views of Afghanistan from around the 10th century onward. The Buddhas of Bamiyan are a good example of the Buddhist presence in Afghanistan prior to this.

The question isn't would Rome had been aware of Buddhist ideas and Buddhism, and the answer is yes, and no.

Prior to the Hellenistic era founded by Alexander the Great, there was limited interaction between Greece and Indian, mainly because the wide number of peoples and the vast distances separating them. Herodotus sites the problem of distance the classical Greek desire to travel when one of the kings of the Lacedaemonians (sorry, I don't have Herodotus in front of me, I don't remember which of the two kings it was) is being asked to march 1500 stades from the sea to attack Sardis, and he's incredulous at being taken that far from the sea.

And the Classical Greeks considered all non-Greek speakers, or non-Greeks as barbarians. The xenophobia the Hellenes exhibited was at times legendary and celebrated.

After Alexander, well, this is a different matter. Pyrrho studied in India and appears to have been influenced sramanic or Buddhist thought in the few sayings of his that survive. "Nothing really exists, but human life is governed by convention."

Hellenistic artists were sought to create Buddha carvings, Indo-Greek and Greek Buddhist missionaries were sent as far was as the Mediterraean by Ashoka. The Greek world knew of India as a place of gymnosophists, where literally "naked philosophers" dwelt and taught philosophy. The Greeks were pretty willing to play with different philosophies, even religious ones. As evidenced with the eastern Greek kingdoms embrace of Buddhism, That is probably the reason right there - Buddhism arrived as several different philosophies and schools, heavily filtered through Greek logic and rhetoric, and when Buddhism the religion arrived, it would have one of many competing philosophies. But it would have been known of, by some people.

εἰσὶ δὲ τῶν Ἰνδῶν οἱ τοῖς Βούττα πειθόμενοι παραγγέλμασιν. ὃν δι’ ὑπερβολὴν σεμνότητος ὡς θεὸν τετιμήκασι. [Among the Indians are those philosophers also who follow the precepts of Boutta, whom they honour as a god on account of his extraordinary sanctity.]

— Clement of Alexandria, Stromata (Miscellanies), Book I, Chapter XV

This suggests that the Romans would have been aware of Buddha, and through the Greeks aware of Indian Philosophies. The Romans traded extensively with India, and at least one Senator was said to have complained that Roman gold, wine and olive oil disappeared into India and all Rome got was silk for it's vanity. (Badly paraphrased from memory.) But this wasn't entirely true, Rome also knew the Indian-Chinese (Parthian) peoples as the makers of the best/second best steel in the world, with actual Chinese (forget the Roman name started with an S) steel being the best overall.

So ultimately Indian philosophy was just one of many different religious and philosophical movements flowing into the Mediterraean.

With regard to the question would citizens of Rome have known about Buddhism ... Princeton theologian Elaine Pagels argues in The Gnostic Gospels that Gnostic Christianity, which developed in Roman Egypt, was heavily influenced by Buddhist philosophy. However, although she makes (IMO) a strong case, this is a highly controversial proposition among theology academics and is by no means universally accepted.