I believe wolves could have started the domestication process. I mean wolf pups have similar (but not identical) behaviors to dogs and any abandoned or orphaned wolf pups would easily become attached to a human as the alpha. Actually, humans are pretty well suited to the alpha role, due to the fact that in wolves the leader is generally the best at leading (not necessarily the strongest or fastest, just like in humans) and the alpha always gives out the food. Early human societies would have likely had the similar behaviors as alpha wolves.
Neanderthals may have had a stronger alpha than human societies do. It's likely never going to be provable unless we clone a ton of Neanderthals and dump them on a small island, but if Neanderthals had a more leader dominant pack, similar to what Silverback Gorillas have, then this would have made perfect pack-leaders for orphaned wolf pups. However, from what I've heard is that the process was only ever started, IE wolves were found to live within human populations with signs of domestication (likely curious docile ones), but it was never completed meaning they're unsure if we bred the docile wolves until around 7000 BC.
The one thing I've always wondered is if Neanderthals simply grasped the beginning of concepts. They adopted tools they never invented, but learnt to make. They possibly began the domestication process, but never completed it. They obviously understood how to make clothes that no other species of hominid ever did, because they lived in the UK and some of their sites are behind the glacial lines for most of the period they were in Europe, so they obviously lived close to the glaciers to establish settlements as the glacier retreated. However, they never appeared to advance far on any of these, whether this is merely a symptom of technological development (IE since humans began it's been on exponential growth, however in the beginning things were on the order of 10,000 years apart) or if it's the limit of the Neanderthals brain structure. A lack in neural connectivity may have inhibited creativity and original ideas.
And since neanderthals were almost exclusively meat eaters, it might be the case that they attracted a lot more wild animals with their leftovers.
Another theory I've heard is that neanderthals are unique in having an inferior sense of balance.
Inferior not just when compared to us, but to almost every known hominid or great ape.
Apparently the inner ear organs responsible for balance and thus also for agility do fossilize and that's how we know.
So part of their extinction might have been our ability to outrun and out jump them.
Then again it could have just been our pernicious desire and ability to exterminate competition.