Is it a bad idea to mention informal PhD offers in a job application? I am about to get my master's degree and have published a few papers in highly reputable computer science journals. I have gotten an "informal offer" for a PhD position from a world-class university. An informal offer here means that a well-funded professor has e-mailed me, strongly encouraging me to apply and to make sure to mention his/her name in the application so it gets routed to him/her. 

For various reasons, I'm leaning towards not pursuing a PhD so I'm currently working on my resume. I'm wondering if it's a bad idea to mention anything about this informal offer in the personal letter. The professor making the offer will not be the one writing my letters of recommendation. Verifying the claim would not be hard, but it's not really a tangible accomplishment. On the other hand, why shouldn't I highlight the fact that I could get into the best graduate program in the world if I wanted to?
tl;dr: No, you should not mention this "offer."

I wouldn't even call this an informal offer. An "informal offer" would be if you applied, and then were told by e.g. email or phone that you are admitted, and that the formal letter of admission will follow shortly. Encouragement to apply, no matter how strong, is not an offer of admission, and doesn't indicate that you would definitely get an offer of admission if you would have applied.

For example, you may sound great according to whatever information you sent to that professor, but your full application may reveal other things that lead the admissions committee to reject your application. Or maybe you really are that great and the professor has funding, but another, better, student applies that year and gets the offer instead of you.

Furthermore, individual professors have varying ability to actually influence the admissions process, depending on how graduate admissions actually works in that department.

Even for a "real" offer of admission, it wouldn't necessarily be a good idea to list it on a résumé. See for example this related question on listing fellowships that are offered, but which you decline. But being encouraged very strongly to apply to something is certainly not a CV-worthy accomplishment.