Logic is one of those things about which most people have more misconceptions than correct ideas. Take the word "reasoning". The relevant definitions are as follows:
Reasoning 1. the act or process of drawing conclusions from facts, evidence, etc.
Reason 2. a. The capacity for logical, rational, and analytic thought;
Reason is the capacity of consciously making sense of things, applying logic, and adapting or justifying practices, institutions, and beliefs based on new or existing information — https://en.wikipedia.org/wiki/Reason
Given these definitions, we may get the impression that typical reasoning consists in a conscious process of drawing conclusions using our capacity for logical, rational, analytic thought. Yet, it turns out that we really suck at doing this. We all have opinions on things, but when we are challenged to justify our claims, we invariably fail to provide anything like a proper reasoning as defined and indeed as usually understood. All we can usually do is exhibit a few paltry reasons, reasons that we usually are again incapable of justifying properly. And this is not just "ordinary folks". Lawyers may be considered as professionally motivated to be good at it, and yet, they too fail. They certainly can string together a large number of reasons, often reasonable enough, but in the end nothing that would constitute any actual logical reasoning. We can verify this also with philosophers, in particular with analytic philosophers, the ones who actually tried really hard to do it. And who also failed. Perhaps the best evidence of our inability to produce reasonings as we usually understand the term is the failure of mathematical logic to formalise the mathematical reasoning supposedly justifying mathematical proofs. The best that mathematicians have been able to achieve in this respect is to adopt the material implication as the most accurate model of the logical implication at the heart of any logical reasoning. Yet, it is apparent, and philosophers and mathematicians who know the history of logic recognise this, the material implication is not any accurate model of the logical implication. And thus, mathematical logic does not provide any formal description of any mathematical reasoning. This does not stop mathematicians doing mathematics and producing proofs, just not anything like formal proofs of their theorems. More explicitly, whatever mathematical logic proposes in lieu of formal proof has in fact little relation to the logical reasoning of mathematicians themselves.
It is the failure of mathematicians to describe logical reasoning which has come to provide the rationale and the motivation now for many philosophers to reconsider the fundamentals of our view of human reasoning. Essentially, most philosophers now seem to have resigned themselves to the idea that there is in fact no such a thing as logical reasoning. It is of course easy to exhibit all sorts of examples of actual human reasoning that seem to confirm this relatively new perspective. And so the conclusion is now that human reasoning is so riddled with cognitive biases and quirks that the search for the logic of human deductive reasoning can only be a goose chase.
Yet, it is easy to falsify this view. For example, it is a fact that Aristotle's syllogisms all seem perfectly logical to us. Similarly, all logical truths discovered by traditional logicians since Aristotle also all seem perfectly logical to us. Mathematicians themselves discovered some logical truths, such as for example De Morgan's Laws. And nobody has any doubt that they are correct (Pierce's Law apart, but this is a special case). The few arguments philosophers have proposed purporting to show that the modus ponens or the modus tollens are not quite true or don't apply to all cases are gross logical fallacies and mostly recognised as such.
There is no doubt that there is such a thing as the logic of human deductive reasoning. It is by definition very limited. The logic of something is by definition the few basic principles that underpin the way something works. Thus, we shouldn't ask formal logic to provide a model of the psychology of human reasoning. Logic is by definition limited to the fundamental principles of deductive reasoning, principles we can easily recognise when we see them. What reasonings people actually do is something else entirely. We are all free to reason illogically and we do it all the time. I don't believe for a moment that we can think illogically, but we can without the shadow of a doubt articulate illogical reasonings. Indeed, it is probably easier to do that than to bother articulate a properly logical reasoning. Even mathematicians themselves never bother to articulate formal proofs of their theorems. They think it is good enough to simply provide the minimal explanation which will be sufficient to satisfy their peers that the proof is good. A sort of very sensible gentlemen's agreement.
So it is indeed apparent that a model of actual human reasoning is beyond our current means of investigation, but this shouldn't lead us to throw the logical baby with the bath water. Logic is here to stay and once you put in perspective the historical failure of mathematical logic, it is easy to see that humans do have a logical capacity which can only be native, and to see that Aristotle provided a very good portraiture of it. Nothing like a formal model yet, but good enough that we can all recognise the thing when we see it: If A and B is true, then A is true, but A and B is true, so A is true.
So I don't believe we are close to producing any good description of the way humans think, but producing such a model will inevitably require that we first produce a formal logic which will be an exact model of the logic of human deductive reasoning. The debate currently is muddled by the failed attempt at mathematical logic, but there is little doubt that we will move beyond this problem at some point. Philosophers have also displayed the limitation of their discursive methodology. Most of the discussions philosophers have on the subject is either a Scholastic exegesis of mathematical logic, which cannot possibly provide any clue about the actual logic of human deductive reasoning, or an effort to find another direction. Yet, the subject is well known and it is properly identified. All we need to do now is to develop a properly empirical science of logic, as opposed to the mathematical science of logic, which was doomed to fail from the start simply because mathematicians are not empirical scientists and so don't have the methodology adapted to the task.