A PSA test looks for proteins associated with prostate cancer and the result was high in Lord Cameron's case.
He told the newspaper he wanted to use his platform to support a call by Prostate Cancer Research, a charity which counts Mr Jones - founder of private members' club chain Soho House - as a trustee, for screening to be offered to high-risk men.
The cancer is most common in older age - particularly among men over 75. Cases in the under-50s are rare. It is also more common in black men.
"I don't particularly like discussing my personal intimate health issues, but I feel I ought to," Lord Cameron said.
"Let's be honest. Men are not very good at talking about their health. We tend to put things off."
But he said: "I sort of thought, well, this has happened to you, and you should lend your voice to it."
Lord Cameron, Conservative prime minister between 2010 and 2016, and later foreign secretary in Rishi Sunak's government, told the Times: "I would feel bad if I didn't come forward and say that I've had this experience.
"I had a scan. It helped me discover something that was wrong. It gave me the chance to deal with it."
There is currently no screening programme for prostate cancer in the UK because of concerns about the accuracy of PSA tests.
A high PSA does not always mean cancer. It can go up if there is an infection, for example.
Currently, men over 50 can request a PSA test, which looks for abnormally high levels of protein in the blood, but this is unreliable, picking up many prostate cancers that would never need treatment, and missing others that do.
Some men with a raised PSA may have a prostate cancer that would not have caused problems or needed treatment, causing unnecessary worry and further investigations.