The Shortcode API states that you simply can't have square brackets inside your attribute. Thus, the next wouldn't work:

[tag attribute="[Some value]"]

During my situation, the square bracket is needed. What will be the best answer of having for this problem? I have already attempted getting away this content during my shortcode function, but didn't have luck.

I'm using WordPress 3.3.1.

Apply certain other special character inside your shortcode and change it square brackets inside your shortcode function.

When the brackets appear included in the HTML produced use &#...; or included in an URL use %... .

Otherwise, whether it concerns your personal shortcode, just replace another character, e.g. {} by [], within the code from the shortcode.

When not your personal shortcode, you may customize the wordpress plugin / core. I'd write a wrapper code so as not to interrupt updates.