Problems with exporting AI files when placed images are involved.

So I have a bit of a problem which I am wandering if can even get worked out?

 

I created a design in Corel X5 that has PSD and TIFF images in it.  The client needs them in an AI file.  When I exported the file the images did not come in correctly they were pixelated and broken up.

 

I think this is a common problem and is there anywhere around it for it to be able to be exported correctly as an AI file.  I really don't want to lose this job/client?