No, it won't boil or condense. It will evaporate faster, but it won't boil. Boiling requires bubbles forming on solid surfaces. In everyday situations, the surfaces are much hotter than the water, in the order of 20°C more.
In the arrangement that is described, this can't be achieved, as the whole body of water (the water and the steam) outside the glass will be at boiling point. There is no way in which the glass can be heated above boiling point.
When the outside body of water is heated more, there are more bubbles which enhances heat transfer a little bit. The water in the glass will evaporate a little bit faster.
Remember that heat transfer requires a temperature difference. If the water put into the glass is at boiling point initially, there won't be any heat transfer and much less boiling. If it is below boiling point initially, it cannot boil.