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  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 . 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.
: Chen, Lu, et al. "Identifying and interpreting apparent Neanderthal ancestry in African individuals." Cell 180.4 (2020): 677-687.
: Sankararaman, Sriram, et al. "The genomic landscape of Neanderthal ancestry in present-day humans." Nature 507.7492 (2014): 354-357.