While I have never heard of any evidence of middle-eastern origin DNA in mesoAmerica, there is a lot of evidence that there have been several other contacts from outside the Americas, in addition to the traditional "Berring land bridge" theory.
There is evidence of human habitation in the Americas as early as 35,000 years ago.
There is DNA genetic evidence of south sea islander culture in South America and even evidence of South American culture affecting Easter Island.
There is the "red paint people" culture that seemingly may have spanned from New Jersey around the northern rim of the Atlantic to Portugal, thousands of years before the Vikings reached North America.
We lost a lot of the maritime archeology that occurred before the 325' sea level increase after the last peak of glaciation, about 25,000 years ago. Any seaside cultural artefacts were demolished as the surf along the coasts slowly advanced through the areas where people would have made fishing villages long ago. But, we do know from some burials that people were fishing for deep water species like swordfish maybe as much as 8,000 years ago in the northeastern part of North America. So, how people were able to travel around the globe thousands of years ago is just now beginning to be appreciated.
Which gets us back to using DNA of current inhabitants to try to understand ancient population interactions. But, as we tend to mix our personal ancestries faster and faster with modern travel opportunities, DNA investigation is increasingly dependent on old burials.
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