The Olmecs flourished around 1200 BC on the southern Gulf coast of Mexico and spread their influence in Central America from Belize to Costa Rica until around 300 BC when they just disappeared!
David Childress wrote an interesting article(c) on what little is known about the origin of the Olmecs in the 2007 Sept/Oct. issue of Atlantis Rising magazine. This was one of a number of promotional pieces(e) for his book The Mystery of the Olmecs published earlier that year. In early 2014, Frank Johnson published a lengthy paper(f) debunking Childress’ Olmec book, with further criticism from Jason Colavito following shortly afterwards(g).
Jacques de Mahieu, the French Nazi, claimed that the Olmecs were descended from refugees that fled from Troy after the Trojan War. He goes further claiming that the Trojans had originally come from Scandinavia!
The Olmecs have been linked by a variety of writers with Atlantis. The first Latin writer of Aztec history was Fernando de Alva Cortes Ixitilxochill, of Aztec lineage, who maintained that the Olmecs had come to Eastern Mexico from the Antilles via Florida.
At the end of the 17th century, a former Jesuit, Don Carlos de Sigüenza y Gongora who had befriended the son of Ixitilxochill was allegedly in possession of a remarkable collection of native manuscripts that had escaped the insane mass burnings of the 15th century. He believed that the earliest inhabitants of Mexico had come from Atlantis. The Olmecs who preceded the Toltecs were not identified until the 19th century.
The 1973 discovery of a grooved 3.5 cm hematite bar by Michael D. Coe at the San Lorenzo site led researchers to immediate conjectural comment. In 1979, Robert Temple wrote two articles(o)(p), for Second Look magazine, on this find and the possibility of Olmec knowledge of magnetism. However, wild speculation led to the conclusion from this single artefact that the Olmecs had invented the compass. This assumption led to the further suggestion(j)(k) that the Olmecs had advanced navigational skills and with a final leap of imagination decided therefore that they had arrived in America from their homeland, Atlantis!
Ivar Zapp & George Erikson use the stone spheres of Costa Rica as the starting point for their controversial book on Atlantis, insisting on a connection between the stones and the Olmecs. Gene Matlock considers the Olmecs to have had Hindu origins(t) and to be possible survivors from Atlantis and Clyde Winters holds similar views and has supported them with his book  on the subject. In the same book[p.13] he offers Libya as the home of Atlantis, while elsewhere Matlock suggested(d) that the Olmecs were Turkish!
However, Afrocentrist, Paul Barton, claims(m) that the Olmecs came from the Mende people of West Africa who are now one of the principal ethnic groups of Sierra Leone. Alessandro Demontis, who is something of a Zecharia Sitchin apologist, has written a short paper(r) on the leading proponents of the concept of an African origin for the Olmecs and argues that the academic background of many of them demands greater consideration of their viewpoint. Demontis thinks that to simply dismiss them as pseudoscientists is unwarranted.
The Negroid features that are clearly to be seen on some of the Olmec carvings are probably evidence of an ancient link with Africa and nothing more. Many think that the Olmec heads are the only evidence for pre-Columbian links with Africa, however, that is far from the truth as dozens if not hundreds of artefacts displaying African features have been discovered in the Americas(u)(v), although I cannot help noting that there is an obvious Asian influence in some of the figures!
On balance, I do not think that any credible case for identifying the Olmecs with Atlantis has yet been made. However, in my opinion, a far more convincing case has been made for linking the Chinese with the Olmecs(a)(b) and in that regard, the book by Dr H. Mike Xu, Origin of the Olmec Civilisation, is worth a read. Similarly, Charlotte Harris Rees has compiled a vast collection of data linking the Chinese with America in her Secret Maps of the Ancient World. She devotes chapter 4 to the Olmecs. Jacques Gossart has also reviewed(l) the evidence for the Chinese ancestry of the Olmecs. R.A. Jairazbhoy proposed in his book Ancient Egyptians and Chinese in America that as well as Chinese, there are also Semitic, Egyptian and African traces to be found among the Olmecs.
Archaeologist Betty J. Meggers (1921-2012), who worked at the Smithsonian Institution has made a study of the apparent cultural similarities between the Chinese Shang dynasty of c.1750 BC and the Olmecs dated to some centuries later. This she did in a lengthy paper in American Anthropologist in 1974(w).
Many Mormons believe that the Olmecs were the Jaredites who are only referred to in their Book of Ether. This idea has been developed in a short Kindle book by John Dreha.
>This multiplicity of suggested origins for the Olmecs is confusing and was made more so by the observations of Bibhu Dev Misra that many Olmec artefacts show figures adopting a range of yogic poses, which adds to the possibility of an Indian cultural influence. His 2017 illustrated paper on Graham Hancock’s website is intruiging(x).<
In 2006 it was announced(a) that a stone slab was discovered in the Mexican state of Vera Cruz, which appears to be the earliest known writing in the Americas and attributed to the Olmecs and dated to around 900 BC.
The most extreme theory regarding the Olmecs that I have found, is the claim that they were astronauts. This idea was expressed(h) by Xavier Séguin, quoting US astronaut Gordon Cooper ! However, I discovered Séguin to be unreliable when I found that he also quoted a fictional character, Professor Mortimer, from a work by the renowned writer, Edgar P. Jacobs, as supposedly uttered by a real scientist(n), concerning the Pillars of Heracles.
Zechariah Sitchin in The Lost Realms  claims that the Olmecs were culture bearers who arrived in America circa 3000 BC, which conflicts with the date of 1500 BC proposed by conventional archaeology. Sitchin also quoted(s) Cooper’s heavily panned book, A Leap of Faith .
In 2006, it was announced(i) that a stone slab was discovered in the Mexican state of Vera Cruz, which appears to be the earliest known writing in the Americas and attributed to the Olmecs and dated to around 900 BC.
May 2017 brought an interesting article(q), on the Ancient Origins website, outlining the unique features of the Olmecs.
(a) See Archive 2532
(e) https://www.transformtheillusion.com/articles/David%20Childress/The%20Mystery%20of%20the%20Origin%20of%20the%20Olmecs%20.html (offline Nov. 2016) see Archive 2294
(f) https://ancientaliensdebunked.com/mystery-solved-olmecs-and-transoceanic-contact/ (offline May 2018) See: Archive 2295
(j) See: Archive 2776
(l) http://www.orbs.fr Issue #5