Is it possible to breed neanderthals through selective breeding?

Selective breeding will select for genetics that are already present in your population, but won't introduce new DNA sequences that aren't carried by any of the population members. It would technically be possible if everyone carried a different 1-4% of the Neanderthal genome, since it would be possible to reconstruct the full Neanderthal genome by creative combinations of DNA that exists in the population. That's not the case, though, it will be a fairly consistent set of regions for which you can substitute Neanderthal DNA for homo sapien DNA and still have a functional human.

So no, this isn't possible - the complete genome of Neanderthals doesn't exist split among the human population, so it can't be refined to be complete through selective breeding.

I agree with the two previous posts, but they are a bit light on references / evidence, so I thought I'd throw my two cents in.

The answer it seems is most likely no. As others have alluded to, in order to 'breed' a Neanderthal selectively from modern humans, it would need to be the cases that the entire ~3.1Gb of the Neanderthal genome would need to be represented in modern humans.

Methods exist which are able to identify the specific segments in modern individuals which are derived from the Neanderthal admixture event. One recent example is IBDmix [1] from Josh Akey's group in Princeton. They inferred 110.98Gb in total across 2504 worldwide individuals from the 1000 genomes reference panel, which works out as an average at about 0.04 Gb per individual, or 1.3% on average across all individuals. There is a substantial amount of variation between population, with Africans having less Neanderthal ancestry than non-Africans.

However, when you collapse down the Neanderthal segments into non-overlapping regions, it turns out we can only identify 1.29 Gb of unique segments. This means that we have 1.29Gb / 3.1Gb = ~40% of the entire Neanderthal genome present in modern day humans. A similar, older study inferred a slightly smaller amount of 1.1Gb [2]. It is very unlikely this figure will change a large amount in the future.

This is why 'breeding' such an individual with 100% Neanderthal DNA would be impossible; around 60% of that ancestry has been totally lost from modern humans, either through genetic drift or purifying selection.

References

[1]: Chen, Lu, et al. "Identifying and interpreting apparent Neanderthal ancestry in African individuals." Cell 180.4 (2020): 677-687.

[2]: Sankararaman, Sriram, et al. "The genomic landscape of Neanderthal ancestry in present-day humans." Nature 507.7492 (2014): 354-357.