Someone asked a very similar question last year. Here you go:
Alexander certainly recognized the importance of maintaining a presence on the Italian Peninsula and how beneficial its position was on the Adriatic. After receiving a request for help from the Greek colony of Tarentum which was being threatened by various Italic tribes, he sent his brother-in-law Alexander of Epirus to provide aid. After Alexander of Epirus had landed, Livy tells us (8.17) that he made peace with the Romans. Not only was Alexander III expanding Macedonian armies throughout Greece and its settlements, but the Macedonian alliance with Rome also allowed Alexander III to address the problem of piracy within the Adriatic. Livy expresses doubt that this treaty would have lasted long had Alexander of Epirus’ “other enterprises been equally successful.” He remained a formidable force in the region until his death during battle.
Returning to Babylon for the winter following Hephaestion’s death, Alexander III’s army wasn’t sure what his new plan would be or where he would take them, so it was entirely up to speculation. (His behavior had grown increasingly outlandish which was probably aggravated tenfold by the death of Hephaestion.) Envoys came from all over the world to secure their positions abroad.
The following is from Robin Lane Fox’s Alexander the Great
“These distant envoys at once raised the issue of what Alexander would do next, an indication of his state of mind. If Carthage, Libya, and Spain promised friendship there was little to stop him marching west through Egypt, along to the Pillar of Hercules, nowadays the Straits of Gibraltar, and skirting up through Spain to Italy where his brother-in-law had lost his life on an expedition. Rumour spread that western conquest was his new ambition; some even suggested that he meant to sail completely round Africa and enter the Mediterranean from the Atlantic…It is just possible that Alexander had considered it, for the rumour of the plan may perhaps go back to Nearchus, with whom he had discussed plans at Kirman in the previous spring.
However, Alexander ended up setting his sights on Arabia for several more enticing reasons. (A pretty humorous one was, according to his officer Aristobulus, “the fact that the Arabs had never sent him an embassy” which is just typical Alexander if you ask me.) As to Livy’s potential knowledge of this, I couldn’t say. I haven’t looked much into the various ancient sources/histories of Alexander as it’s a murky, murky swamp I haven’t yet had the mental energy to wade, but I do know that most—if not all—works by his original historians are only second or third hand accounts, and I don’t know how available these were to Livy. Sorry I couldn’t be of more help on that front!
The Greek historian Arrian, writing in the 2nd century AD, claims that before his death Alexander was planning his next conquest to be the Arabian Peninsula and the islands of the Persian Gulf, such as Bahrain. If Arrian is to be believed, Alexander was in the process of building up a large fleet for the purpose of the invasion. Arrian was writing several centuries after Alexander's death, of course, but he had at his disposal many works written by people who were close to Alexander, works that haven't survived to the present day. Arrian's history on Alexander, Anabasis Alexandri (The Campaigns of Alexander), is well-regarded and is probably the best ancient source on Alexander the Great.
From Book 7, chapter 19 of Anabasis of Alexander:
For Alexander designed to colonize the sea-board near the Persian Gulf, as well as the islands in that sea. For he thought that this land would become no less prosperous than Phoenicia. He made these preparations of the fleet to attack the main body of the Arabs, under the pretext that they were the only barbarians of this region who had not sent an embassy to him or done anything else becoming their position and showing respect to him. But the truth was, as it seems to me, that Alexander was insatiably ambitious of acquiring fresh territory.
https://archive.org/details/cu31924026460752
The 1st century BC Greek historian Diodorus Siculus also wrote about Alexander as a part of his monumental Bibliotheca Historica (Historical Library). In this, he claims that Alexander left a will to his general Craterus with a series of designs that he wanted his men to carry out. Among other things (like building a tomb for Philip II that would rival the pyramids of Egypt), this included the conquest of Carthage the other regions along the coast of Libya (which to the Greeks meant North Africa west of Egypt), as well as the northern Mediterranean coast from the Iberian Peninsula (Spain) to Sicily. If Alexander's successors actually carried out this plan in its entirety, it would have likely meant running into Rome. According to Diodorus, Alexander's successors deemed these plans unrealistic and ignored them.
When it comes to Alexander, though, Diodorus (despite writing closer to Alexander's time) is less specific (perhaps due to the nature of his work, which is a general history) and likely less reliable than Arrian. Alexander undoubtably had huge ambitions, though, so he might have very well desired to conquer the Mediterranean. Still, this should all be taken with a grain of salt. Arrian doesn't mention this and Arrian is probably the best ancient source on Alexander, as aforementioned.
From Book 18, chapter 4 of Bibliotheca Historica:
The following were the largest and most remarkable items of the memoranda. It was proposed to build a thousand warships, larger than triremes, in Phoenicia, Syria, Cilicia, and Cyprus for the campaign against the Carthaginians and the others who live along the coast of Libya and Iberia and the adjoining coastal region as far as Sicily...When these memoranda had been read, the Macedonians, although they applauded the name of Alexander, nevertheless saw that the projects were extravagant and impracticable and decided to carry out none of those that have been mentioned.
http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Diodorus_Siculus/18A*.html#3
So, the idea that Alexander might have turned to the Mediterranean is something that was commonly speculated about. I believe Plutarch talks about it as well. Livy may have very well been going off of works such as Bibliotheca Historica and whatnot. Livy didn't think too highly of the idea that Alexander could have conquered Rome. Livy actually goes on a bit of a rant in chapters 16 through 19 in Book 9 of Ab Urbe Condita Libri (The History of Rome). It's really quite entertaining.
From Book 9, chapter 19 of The History of Rome:
He [Alexander] would have crossed the sea with his veteran Macedonians, amounting to no more than thirty thousand infantry and four thousand horse, these mostly Thessalians. This was the whole of his strength. Had he brought with him Persians and Indians, and those other nations, it would be dragging after him an encumbrance rather than a support. Add to this, that the Romans, being at home, would have had recruits at hand: Alexander, waging war in a foreign country, would have found his army worn out with long service, as happened afterwards to Hannibal. As to arms, theirs were a buckler and long spears; those of the Romans, a shield, which covered the body more effectually, and a javelin, a much more forcible weapon than the spear, either in throwing or striking. The soldiers, on both sides, were used to steady combat, and to preserve their ranks. But the Macedonian phalanx was unapt for motion, and composed of similar parts throughout: the Roman line less compact, consisting of several various parts, was easily divided as occasion required, and as easily conjoined. Then what soldier is comparable to the Roman in the throwing up of works? who better calculated to endure fatigue? Alexander, if overcome in one battle, would have been overcome in war. The Roman, whom Claudium, whom Cannae, did not crush, what line of battle could crush?
http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.02.0147%3Abook%3D9%3Achapter%3D19
So, basically, I'd say that there is not enough evidence to definitively say that Alexander was actually planning on conquering Italy, but many people thought he would have attempted to expand into the Mediterranean and Livy would have been influenced by those beliefs (well, influenced enough to go on a rant about how Rome could totally kick Alexander's butt).