Is it self-plagiarism to summarise a topic in the introduction of two of your papers using similar references and examples? I have written two papers, let's say X and Y, about a concept, let's say A. 

X is about other overview of concept A and how it can be improved in general. 
Then I have another paper i.e. Y discusses about relationship of A with reference to one particular topic. 

As both papers are literature driven papers (can be called as literature reviews) and the focuses are different. However there are some sections which are overlapping with each other. Therefore I have given similar examples and used many common references. 

My question is, is it a case of plagiarism ? I have written both papers separately and not copied text of one into other, rather sometimes the sense is same. I would appreciate your thinking and advices in this regards whether it can be counted as a case of plagiarism ? 

Thanks in advance.
As you have stated, your papers have similarities and that the text is written separately and not copied, then no, this is not plagiarism (and is unlikely to be seen as such).  Using the same references is fine and necessary for similar concepts. 

It also sounds like the context between the two papers are distinct, this should ensure that the two papers are written differently.

Make sure you re-read and compare them to ensure that they are sufficiently different, which you sound like you have already done (and possibly ask a colleague to compare them). If possible and relevant, you could have one of your papers cite the other.

There are many guides about plagiarism and what to check for online, for example, this one from the University of Western Australia.