This hypothesis is based on one and one ancient source only, and it's a passage that comes from Herodotus' Histories, which was written around 440BC. When describing the geography of Africa, he tells about a circumnavigation project that by his time would have already been old history. Necho II, who supposedly ordered Phoenicians to sail around Libya (i.e. continent of Africa) because of his defense projects, ruled in 610-595 BC. We have no way of knowing what's Herodotus' source for this story - since Herodotus did not have the language skills to read African written records, he's most likely reporting a story that's been passed down to him orally. Here it is:
Libya is washed on all sides by the sea except where it joins Asia, as was first demonstrated, so far as our knowledge goes, by the Egyptian king Neco, who, after calling off the construction of the canal between the Nile and the Arabian gulf, sent out a fleet manned by a Phoenician crew with orders to sail west about and return to Egypt and the Mediterranean by way of the Straits of Gibraltar. The Phoenicians sailed from the Arabian gulf into the southern ocean, and every autumn put in at some convenient spot on the Libyan coast, sowed a patch of ground, and waited for next year's harvest. Then, having got in their grain, they put to sea again, and after two full years rounded the Pillars of Heracles in the course of the third, and returned to Egypt. These men made a statement which I do not myself believe, though others may, to the effect that as they sailed on a westerly course round the southern end of Libya, they had the sun on their right - to northward of them. This is how Libya was first discovered by sea.
Herodotus, Histories 4.42; tr.Aubrey de Sélincourt.
Although Herodotus does not believe that sun would behave as strangely as this, this is indeed where the sun is if you sail, past the Tropic of Capricorn, towards and along the southern coast of Africa. Necho's men thus would have sailed clockwise around the continent, starting from Egyptian coast of the Red Sea, down the east coast, up the west coast, and to the Mediterranean through the Straits of Gibraltar (Herodotus' 'Pillars of Heracles'). Historians have actually confirmed that this journey would have been possible with the 7th-6th century BC ships, and even considered the different stages of the journey, the winds, and the currents, and concluded that it would have been entirely possible and could have indeed taken about two years, and ran well into the third if you consider all the possible stops and rests, waiting for favorable winds etc. and other pauses they were forced to take during their trip. Livius has quite a detailed summary of a potential reconstruction of their journey. You'll find academics who find Herodotus' story as conclusive proof that the Phoenicians' journey did indeed happen - Ciaran Branigan seems to be one of them.
That said, although it is perfectly possible that the journey was actually made, Herodotus' story by no means confirms that it took place and I'm personally pretty sceptic about it. Herodotus' says that 'Libya is surrounded by sea on all sides', but he would not have needed some scientific confirmation for this: he, as almost any of his contemporaries, would have thought this to be the case, since the classical Greeks believed that the earth was surrounded by sea on all sides: see map of the world of Herodotus. And, although Herodotus himself does not seem to be very well versed on this field - as intelligent and well-informed as he was for his own time - there were Greeks who knew perfectly well that the world was round (this discovery is usually attributed to Pythagoras, c. 570-495 BC) and who understood the movements of the sun.
Here's A. Lloyd's reconstruction of the contemporary Greek knowledge of the relationship between the globe and sun's movements, and the route of the alleged circumnavigation. If a Greek were told of an expedition that was to pass around Africa, he would have expected that the voyage took the ship beyond the line of αἲ χειμεριναὶ τροπαὶ, providing that they went far enough south, and they could have inferred that the sun would have been on the right side of the ship. It's actually possible that Herodotus was very well aware of this, and he's not incredulous about the movements of the sun but disbelieves that Libya would stretch out so far south that the sailors passed the αἲ χειμεριναὶ τροπαὶ. Therefore, there's no reason to think that the journey must have taken place. In fact it sounds a bit more like a fabulous rumour, so characteristic of Herodotus, "Listen! Apparently Libya is SOOO long, that if you sail to the end of it, you pass the αἲ χειμεριναὶ τροπαὶ!".
There's a couple of further historical reasons to be sceptic about the journey, and Alan Lloyd summarized them well (The Journal of Egyptian Archaeology, Vol. 63 (1977), pp. 142-155). Firstly, Necho's solution to his alleged problem seems extremely irrational. Herodotus' characterization of Necho is somewhat anachronistic, portraying him as some sort of 'philosopher king' who acts quite contrary from the customary ancient Egyptian thought-process. If an Egyptian king, at any period, organized and dispatched an expedition, he did it usually with specific practical needs and for specific practical ends, and not to satisfy his curiosity. Obviously, Necho might have been an exception, but he would have struggled to justify to his contemporaries why to waste ships (which were extremely expensive) and good men on a random project with unpredictable outcomes. Also, all of the sailors would have needed to have an equally 'modern' mindset not to simply give up and turn around to go back home when the going got rough. Secondly, if the journey really did happen, it's somewhat strange that there's not any other sources for it, and why the journey completely faded from the popular memory and ancient knowledge. Ptolemy's account of the world, which contained all of the Western knowledge of geography from c. 150 AD, considered Africa as an endless Southern land mass. Maps based on it were used all the way up to the Age of Discovery, until the knowledge of the spherical nature of the Earth took root and the Portuguese started actively looking for a way to circumnavigate Africa in order to find a naval route to India. This proved a challenge even to the renaissance naval technology; after a few failed Portuguese attempts, Bartolomeu Dias finally reached the Cape of Good Hope in 1488 but turned back, and Vasco da Gama was the first one to actually navigate around Africa to India in 1497-1499.
TL;DR: It is possible that the journey took place but personally, I find it highly unlikely.
