This actually sounded a lot like Henry Kuttner 's "By These Presents" - a man makes a deal with the devil: make me immortal from everything except suicide, & you can have my soul if I ever do commit suicide!
Here we have a couple of robots - one in the role of Kuttner's devil, another in the role of Kuttner's human. End & motivations, of course, are different from Kuttner's story.
Fact sheet.First published: AntiSF, #139 (January 2010).
Download full text from publisher's site.