Final answer:
No current evidence indicates gene flow between Neandertals and anatomically modern Homo sapiens in the Near East. This conclusion is based on genetic and archaeological research. Although Neandertal DNA is present in modern populations and instances of interbreeding in other regions are well-documented, the interactions in the Near East are not yet clear.
Step-by-step explanation:
There is no evidence from genetic or archaeological research for gene flow between Neandertals and anatomically modern Homo sapiens (AMHS) in the Near East. Recent genetic analyses have revealed varied results in regard to the extent of interbreeding between Neandertals and modern humans in different parts of the world. While it has been established that there was a notable percentage of Neandertal DNA in modern European and Middle Eastern populations, implying some instances of interbreeding, the specific interactions in the Near East remain less clear.
Additionally, findings like the Denisova 11 specimen show evidence of interbreeding between different Homo species, like Neandertals and Denisovans. However, this particular evidence comes from a site in Siberia, not the Near East. In the context of the Near East, archaeology and ancient DNA (Adna) research are still relatively new, so future discoveries could yet provide evidence of interbreeding in this region.
Conclusively, while there is clear evidence of gene flow between Neandertals and AMHS in some regions, no such evidence has emerged from the Near East, maintaining the unclear picture of Neandertal interaction with modern humans in this area.