Reports suggest Karen met Elvis in 1973 and 1975. Certainly at the height of her popularity as part of Carpenters.
Petula Clark, a beloved singer, recalls a meeting with the three of them where Elvis clearly wanted to take the relationship to a sexual level. Whether this is true or only her hunch is not to be known, but it does make for an interesting story.
Corrections as of August 9, 2015:
The top photo has been discovered to be a fraud. The original is below: