Sorry for the late answer, but I hope you can use it anyway!
I don't think there's really any evidence, except for some liberties taken by the screenwriters in portraying an epic tale in the viewers mind of 'What could've been...'. Rome, at the time of Alexander, was busy clashing with the Etruscans and the Samnites, and they weren't really as large a force as they came to be in later years, and they weren't much of a threat to Alexander. Rome did later conquer all the Italic-Greek cities on the coast, what was known as Magna Graecia, but most of this happened after Alexander had died. Alexander's relative, Pyrrhus of Epirus, did fight Rome though, and won a few very costly victories, giving name to the term 'Pyrrhic victory' - a victory won through heavy losses, unable to be sustained. Livy, a Roman historian, did however write a fictional 'What-if' (Kinda) of a potential one-on-one with Alexander vs. the Roman Empire at that time, you can read about that here.
He did have plans and made preparations to invade the Arabian peninsula however, and supposedly had plans to take the Western portion of the Mediterranean - this might be where the writers took some liberties from, and they may have just added some spurious details - I'll talk about this more below. In any case, Alexander marched East, away from Rome, rather than West, towards it. After later conquering the entirety of Asia, reaching as far East as modern Pakistan, he had decided to accept defeat at facing the river Hyphasis - though the 70 days of continuous monsoon, and a brewing mutiny in his camp certainly didn't help things. He decided to march back home, but not through the newly conquered Eastern empire, but southwards down the Indus to the Southern Ocean on a flotilla. He was determined to explore the ocean. On this trip back home, he almost lost his life actually, his armor was penetrated by an Indian arrow when he was isolated on a city wall. They eventually make it back home, and his boyhood friend, Nearchus, had brought his fleet back intact without losing a single ship. Alexander "... began dreaming of future voyages, a circumnavigation of Africa and perhaps even the conquest of the western Mediterranean" ^(1)
After his return home, he first returned to Susa, and later moved on through Mesopotamia to Babylon where he would rest for the summer. In Babylon, he began to plan an invasion of Arabia. The riches of Arabia had become the stuff of legends, and his own reconnaissance of the peninsula suggested he'd be able to settle cities there, like he had previously done on his campaigns through the East. He ordered a harbor to be dredged out of the Euphrates so he could mobilize a navy, and he began gathering men from the vast empire he had conquered. It was rumored, at this time, that once Alexander had conquered the peninsula (Which was apparently a given), he was to turn west in to the Mediterranean, and his focus would be on conquering the rest of the countries and cities nestled in the western part of Alexander's home region. At the news of this, "... a stream of embassies, Greece, Etruria, Libya, Carthage, and even, it was said, Spain made their way to Babylon to offer reverence.". ^(2)
The frantic buzz of activity came to a sudden end in May/June of 323. Alexander, after a heavy night of drinking with his companions, supposedly in one final bout, drank from a bowl able to contain 12 pints and either promptly collapsed dead (If you believe the poison theory), or lingered on for several days and died in June. Some say he was poisoned, some say he was ill or died of the alcohol consumption - either way, by June 10th, he was dead, and his empire began to shatter around him as his generals rushed to grab whatever parts of the crumbling empire they could for themselves.
So, to conclude, I can't find any sources that mention this specifically - maybe the writers took what Livy wrote and ran with it, or they might have confused Alexander the Great with his uncle, Alexander of Molossis (Alexander the Great's father, Philip II, was assassinated at Alexander of Molossis' wedding!), who did invade parts of Italy? I don't know. If you'd like some further reading, I can definitely suggest some, but I don't think I can direct you to reading that'll answer your question more thoroughly. Either way, movies like Alexander (2004), and Troy (2004 & 2018), as well as 300 (2006) should all be taken with a large, heaping bowl of salt. It's Hollywood, not Harvard. ;)
Sources:
Livy, The History of Rome (Book 9), translated by Rev. Canon Roberts. New York: E. P. Dutton and Co. 1912
Freeman, Charles. 2014. Egypt, Greece, & Rome: Civilizations of the Ancient Mediterranean, third edition. Oxford: Oxford University Press
Footnotes:
Freeman, Charles. 2014. Egypt, Greece, & Rome: Civilizations of the Ancient Mediterranean, third edition. Oxford: Oxford University Press. pp. 326
Freeman, Charles. 2014. Egypt, Greece, & Rome: Civilizations of the Ancient Mediterranean, third edition. Oxford: Oxford University Press. pp. 328