Correct use of "being"

I think the wording that you want is "...in the literature, making it possible to..."

The meaning is that "finding the reports" caused or enabled "establishing the link" to be possible.

To make something possible.

EDIT:

If the second clause should cause the first one, you could just use "because" or since:

"similar reports were found in the literature, since it was possible to establish a link among the three definitions"

I understand why you would want to use "being" here, but in english it sounds a bit confusing, it's hard to be sure what the phrase means with being.

Another possibility: "similar reports were found in the literature, considering it was possible to establish a link among the three definitions"

Hope that helps

The problem there is that it doesn't really make sense in English. I mean, you might be able to find a way to parse it that makes it seem grammatically valid, but if so it wouldn't be semantically valid.

Take "being possible to establish a link..." as a phrase on its own. There's a verb, in the progressive, and an object in the form of "possible to establish a link...". The question is, what is "possible to establish a link". That sort of phrase can't just be applied any old thing, because "possible to..." is actually saying that "establishing a link..." is possible. I'm not sure there's any suitable subject in that construction other than "it". "It being possible to..." is a very common phrasing, so one might insert "it" right into your sentence. That would produce the meaning that the fact it is possible to link the definitions is what allows you to find similar reports. In that case, the meaning might be clearer with "given that it is possible..." rather than "it being possible...". In both cases, it is establishing a condition for the first part of the sentence.

If, on the other hand, you mean that the similar reports are what makes it possible to establish a link among the definitions, you might phrase it as "making it possible to...".

As it is, not only does the sentence come across as clearly wrong to a native speaker, it is also not possible to work out what you mean by it.

As to the example in your comment, "it being..." is what you want there. Oh, and "both" should, in that case, be "two", or "the both" should be "them".