Is it permissible to ask the advisor if she has funding to support me as a postdoc for a year? I am about to defend my thesis and I have not yet secured a postdoc position elsewhere. I want to ask my PhD advisor if she can support me as a postdoc till I secure a position elsewhere. Usually in other groups, it is the advisor who offers a position to their performing students. In my group, graduates have always secured position and have moved away so no one had to have this conversation with the advisor. She has not offered me any position till date. 

I have a good rapport with my advisor, but I don't know how to phrase the request so that it doesn't sound awkward and desperate? I am feeling very hesitant in approaching her with this request but I don't see any other options.

I constantly fear that, probably she didn't like me as a student and wants to get rid of me asap. So, that's the reason she has not offered me a position yet.
Regarding the main question, I refer to xLeitix' excellent answer to a similar question.

Addressing the specific parts of your situation, I see two likely reasons why your advisor hasn't offered you anything yet:


publication record - in most fields, securing a post-doc position requires a reasonably good track record in publications. If your publication output during your PhD was not on par, she may be worried if an academic career is for you.
lack of funding - to offer you a position, your advisor would need to have funding for you readily available. This would be most likely if she recently won a grant. You should ask her about that. If she has no funding for you now, you can ask if she knows any other PIs who are looking for post-docs, which might be more likely. 


You can also consider writing your own grant proposal for a post-doc position; some countries have dedicated funding options for that. (However, the decision process may take half a year or more.)