Does being an older PhD applicant for U.S. universities mean there is less chance to be accepted with funding? I have heard this idea many times.

So, is there any truth to it?

Does an older applicant, say, 10 years older than the typical undergraduate, with equally impressive qualifications, get accepted to PhD positions in American universities without funding?  

If so, what is the reasoning behind this decision?

If this idea has some truth to it, wouldn't it be age discrimination?
In my experience as a grad program director, there is zero truth to this rumor. Funding decisions have nothing to do with the applicant's age. Nor is having worked outside of academia for years a disadvantage -- we often look very favorably at students who bring with them substantial life experience beyond a college education.