No,it's not possible at all. A template that is not responsive can't be turned into a responsive one.
The easiest approach that you can find to have a mobile and responsive site is that offered by Responsivizer. By the way your desktop template will be replaced when mobile devices are detected.
The optimized template framework that is embed into Responsivizer will be used when mobile devices are detected, a set of tools will make it possible transition your desktop look and feel also to mobile layout in few easy steps.