It may be possible to use barrel for this, if the 'center' of the distortion is set to somewhere below the rectangle. I even provided the means by which the center can be specified. However I have not played with offset-centered barrel, and do not know just what can be done, and what effects the 'normalized radius' would have on the overall result.
It may be that exactly what normalized radius is used may need tweeking to get something more usable. I know for a fact that in on of my 'reflection' examples, I found that using a different aspect ratio image had marked impact on the results as the image height changed when the width didn't, thus effecting the barrel 'normalized radius' input.
For the example I talk about see...
http://www.imagemagick.org/Usage/advanced/#reflections
Specifically

ASIDE: this image actually breaks the 'rules about reflections' but looked too cool to not include it in reflections, as it had come up a number of times in previous discussions.
It may be that you would need to add a 'fake and temporary extension' below the main image to achieve the desired (or other interesting) results. If someone does try this, please publish your results in this forum, if not this discussion, as it would be of interest to many people.