Theologically, I would say that it wouldn't be possible that a human clone not have a soul. The breath of life (everything we define as a life form) is given by God to all life. This includes everything from humanity down to protozoa. If it's alive, it has the breath of life. This is because God is the author of life and He is the only author of life. And the breath of life makes one "a living soul". (Genesis 2:7)
"Natural clones" do exist. They are called identical twins (triplets, quads etc.). Though even at that, identical twins don't exactly have the same DNA. There are some subtle genetic difference between identical twins.
So any viable clone that's born, because it is a life form; would be deserving of the same respect we would give other lifeforms in it's category. Whether it's a cloned plant, animal or human.
Personally I believe every life form has a soul. They just don't all have the same types of souls. Plant "souls" aren't the same as animal "souls", aren't the same as human "souls". They all vary by capacity to consider their own actions in the environment; as those actions relate to their accountability to God.
I also believe all life has some form of awareness of God. But again, that awareness is dependent on the type of life form. Since a tree can't "think"; it can't consider it's purpose. But make no mistake, plants respond to the presence of other life forms as well as the presence of threats (like fires). So there is a demonstrable form of "intelligence" there.
Animals, having more complex brains though bear the capacity to "connect" to other animals as well as humans. Animals learn from each other and they also learn from man. Domesticated animals and animals in captivity have different behavior patterns than those who have limited or no exposure to humans. Thus we recognize human emotional traits like empathy and even gratitude in animal behavior.
There was an experiment that researchers did with monkeys that was very interesting. One group of monkeys only got "monkey chow" while the other got fresh fruit in addition to monkey chow. These monkeys were in cages where they could see each other. Well once the monkeys that were only getting monkey chow realized the other monkeys were getting fresh fruit; they got mad. They wanted the fruit too. It wasn't "fair" that the other monkeys were getting fruit and they weren't.
So even other traits that we only associate with humans can be seen in animals. And that recognition I believe is because all life possess the breath of life. And because the breath of life comes from God, that is the thread we all have in common. Thus if you've ever had dogs, cats, horses, cows, pigs, even rodents and reptiles, they all have different and varying degrees of "personality". And this is because all life is not "static". It is individual and it is adaptable and that's the result of life being created by God.