Possibly.
Every source I have read on the subject says Neanderthal “emerged” from Africa, but there seems to be scant fossil and DNA evidence to back that up.
As far as I know, no Neanderthal fossils have ever been found in sub-Saharan Africa, and only a few tribes in that region carry Neanderthal DNA.
I have read that Neanderthal fossils have been found in North Africa, but I've never seen them dated.
In sharp contrast, the fossil and DNA record of Neanderthals in Asia and Europe is very large.
From my own reading, I think a good case can be made that Neanderthal evolved from Homo Erectus in southwest Asia about 500,000 years ago, and that all the Neanderthal fossils and DNA in Africa arrived from the outside.
"Like other humans, Neanderthals originated in Africa but migrated to Eurasia long before other humans did. Neanderthals lived across Eurasia, as far north and west as the Britain, through part of the Middle East, to Uzbekistan."https://www.livescience.com/28036-neanderthals-facts-about-our-extinct-human-relatives.html
All very interesting, but the fossil record is still too sketchy.