Funnily enough, in terms of distance from events, Diogenes Laertios is not as bad as you might think. Diogenes Laertios wrote his Lives and Opinions of Eminent Philosophers some time in the early 3rd century AD. Of the principal narrative sources for the life of Alexander III of Macedon, often styled 'The Great', the two generally considered more reliable, the Life of Alexander by Plutarch and the Anabasis of Alexander by Arrian, are generally dated to the reigns of Trajan and Hadrian respectively, in the first decades of the 2nd century AD. The somewhat fanciful account of Quintus Curtius Rufus is roughly mid-1st century AD, while the somewhat maligned (though more recently somewhat rehabilitated) account of Diodoros of Sicily (aka Diodorus Siculus) dates to around 60-30 BC. Given that Alexander was active during the later 4th century BC, that means our earliest extant narrative (Diodoros) was written nearly three centuries later, and the ones conventionally considered the best (Arrian and Plutarch) were written around 420-460 years after the fact. Proportionally speaking, Diogenes Laertios' roughly 550 years of removal isn't that big of a leap.

Indeed, even by the time even Diodoros was writing, most of the core literary tradition around Alexander had more or less consolidated. The particular story of Alexander and Diogenes in fact first appears not in Diogenes Laertios' work, but as an anecdote in Cicero's Tusculan Disputations (5.32), written c. 45 BC:

But Diogenes took a greater liberty, like a Cynic, when Alexander asked him if he wanted anything: “Just at present,” said he, “I wish that you would stand a little out of the line between me and the sun,” for Alexander was hindering him from sunning himself. And, indeed, this very man used to maintain how much he surpassed the Persian king in his manner of life and fortune; for that he himself was in want of nothing, while the other never had enough; and that he had no inclination for those pleasures of which the other could never get enough to satisfy himself; and that the other could never obtain his.

The account of Arrian is much the same (from Anabasis 7.2.1):

So too he is said to have expressed surprise at Diogenes of Sinope, when he found him on the Isthmus [of Corinth] lying in the sun; he halted with the hypaspists and infantry Companions and asked if he needed anything. Diogenes answered that he needed nothing else, but told him and his followers to stand out of the sunlight.

Neither Arrian nor Cicero give any sense of when this meeting may have taken place – Cicero is not writing a work of history, merely citing a historical example; while Arrian's mention of Diogenes takes the form of a flashback in the book covering Alexander's final year (324/3) and is similarly undated. However, in Plutarch's Life of Alexander, we get the anecdote in its expanded form, and with some means of dating it (apologies for the rather flowery translation by Perrin):

14 (1) And now a general assembly of the Greeks was held at the Isthmus [of Corinth], where a vote was passed to make an expedition against Persia with Alexander, and he was proclaimed their leader. (2) Thereupon many statesmen and philosophers came to him with their congratulations, and he expected that Diogenes of Sinope also, who was tarrying in Corinth, would do likewise. (3) But since that philosopher took not the slightest notice of Alexander, and continued to enjoy his leisure in the suburb Craneion, Alexander went in person to see him; and he found him lying in the sun. (4) Diogenes raised himself up a little when he saw so many persons coming towards him, and fixed his eyes upon Alexander. And when that monarch addressed him with greetings, and asked if he wanted anything, "Yes," said Diogenes, "stand a little out of my sun." (5) It is said that Alexander was so struck by this, and admired so much the haughtiness and grandeur of the man who had nothing but scorn for him, that he said to his followers, who were laughing and jesting about the philosopher as they went away, "But verily, if I were not Alexander, I would be Diogenes."

'Now' in this case means some time in the winter of 335/4, shortly after the razing of Thebes by Alexander around the tail end of 335. There is a slight problem though. This does not fit the other chronologies we have. While this may seem peripheral to whether Alexander met Diogenes, so long as it happened to be at an assembly in Corinth, the timing of this assembly nevertheless has a degree of contextual importance, because the sack of Thebes seems to have been regarded by contemporaries as a particularly severe act, and one that more or less terrified most of the Greek states into affirming submission, if temporarily. In other words, when Alexander is supposed to have met Diogenes, had he already committed one of his most infamous atrocities or not? To explain the chronological problem, we need to orient ourselves around the events we can date with reasonable certainty, based on the more chronologically precise accounts of Arrian and Diodoros:

• Late 336 (roughly October; during the archonship of Pythodelos): death of Philip II.
• Late 336/Early 335 (Arrian) or c. July 335 (Diodoros): Alexander makes a show of force in Greece.*
• Summer-Autumn 335: Alexander campaigns against the Thracians, Illyrians and Triballi in the Balkans and on the Danube.
• Winter 335: Thebes declares war against Alexander and is sacked by the Macedonians.
• Spring 334: Alexander crosses the Hellespont into Asia.

* (Arrian's vague wording suggests this was in 336, but this does not fit the timescales provided. Diodoros, who dates these events to early in the archonship of Euaenetos at Athens, is more precise and probably more reliable.)

Plutarch places this assembly at Corinth, at which Alexander was proclaimed hegemon of the Greeks, between the sack of Thebes and the crossing into Asia, placing it firmly in winter 335/4. Arrian, however, writes that Alexander marched down to the Peloponnese to secure his proclamation as hegemon as soon as he ascended the throne in 336, as part of a general show of force which led to the Athenians backing down from their threats of war against Macedon (Anabasis 1.1.1-3), then turned north to fight in the Balkans (1.1.4-6.11). Later in Arrian's account, having sacked Thebes, Alexander again made a deal with the Athenians – and the Athenians specifically – and returned straight to Macedonia (1.10.1-11.1). In other words, we have two irreconcilable chronologies from our otherwise most reliable sources.

Does Diodoros clear things up? Yes and no. What he says is that during the archonship of Euaenetos (beginning in July 335), Alexander made a brief show of force in Greece to forestall the possible forming of an anti-Macedonian coalition, an event which entailed the reception of Athenian envoys (which Diodoros elaborates here far beyond what Arrian does), then got himself elected hegemon at an assembly in Corinth, then marched north again to campaign in the Balkans (17.2.1-4.9). His account of the specifics of the siege of Thebes differs heavily from Arrian's, but in terms of the general chronological points – Thebes declaring war, the city being sacked, the negotiations with envoys from Athens, and the march back to Macedonia – the two accounts correspond.

From this, the most reasonable approach would be to accept the chronology of Diodoros: Alexander spent the first half a year or so of his reign (i.e. mid-autumn 336 to early summer 335) dealing with the political fallout of his father's murder and heading off potential dynastic rivals; then marched into Greece in mid-summer 335 to secure his broader geopolitical situation, as part of which he held the assembly at Corinth; he then campaigned in the Balkans and on the Danube until some time in autumn 335; returned to Greece to sack Thebes in the winter; then continued preparations for invading Asia Minor in the spring. So Plutarch is a bit off: if Alexander and Diogenes indeed met at Corinth, then they would have done so not in the winter of 335/4, when Alexander had proved himself all but a war criminal, but rather in the summer of 335, when he was still a bit of a sabre-rattler but not yet annihilating major Greek communities.

P. R. Bosman's 2007 article on Alexander and Diogenes, the only relatively recent (by that I mean 21st century) scholarship I could find, points out that although traditionally, the likelihood of Alexander making time for meeting Diogenes in the year and a half of his reign that he spent in Greece has been considered quite low, it is by no means implausible especially if we believe Plutarch that it took place during the assembly at Corinth, which would definitely be a situation in which Alexander would be able to wander about a major Greek city for a bit.

On the matter of Alexander's interest in philosophy, that certainly existed: Alexander's court included many philosophers, largely Greek but also later some Indians, who in all the surviving literary accounts (except that of Diodoros) play a significant role in Alexander's attempts to cultivate an ideology of kingship. It's important not to overstate Alexander's philosophical background, however. Plutarch in particular, in constructing an image of Alexander as a Panhellenist and a paragon of Greek culture, really harps on Alexander's connections to Aristotle, but Aristotle only appears twice in Arrian's Anabasis. On neither occasion is he identified as Alexander's teacher, but rather that of Kallisthenes, a philosopher in Alexander's entourage who had argued against demanding proskynesis (obeisance) from the Macedonians and who as a result was later framed as being involved in an anti-Alexander plot and executed.

All this to say it's probably fair to say it could have happened, and we have a pretty reasonable candidate for when and where. The question now is, did it happen, and how?