Sure, it depends on `xclip`, which you might need to install, and `perl` which you probably already have. I only run it while I'm actually copying and pasting as I don't like leaving loops running, that said, with the 1 second sleep it's really not so bad.

  #!/bin/bash
  while true ; do xclip -sel primary -o | perl -pe 'chomp if eof' | xclip -sel primary ; sleep 1 ; done

Note that this affects only the primary selection, e.g. the copy-on-highlight and paste-on-middle-click clipboard. I hardly ever use the rodent, but when I do, this is what I use it for. You could easily adapt the script to the Ctrl-C Ctrl-V clipboard.