About a third of this group carry the bacterium Helicobacter pylori, which makes stomach bleeds three times more likely - and antibiotics eradicate it.
Research has shown taking low-doses of aspirin can cut the risk of cancer.
Professor Jack Cuzick said screening would make the choice to take it a "no-brainer".
University of London epidemiology Prof Cuzick told the BBC's Newsnight programme: "The test is cheap and very easy to do, and eradication takes only five days.
"Bleeding is the only major setback.
"It's trying to identify those who are infected that matters."
The society working with an international team of experts on cancer prevention is expected to publish a statement on the risks and benefits of long-term aspirin use within weeks.
"We will say this looks very important and needs to be further evaluated", Prof Cuzick said.
The society first looked into aspirin as a cancer-prevention measure in 2009, and has reconvened as evidence of potential benefits has grown.
Taking low-dose aspirin for five years halves the risk of developing colon cancer, according to data published two years ago by Peter Rothwell, from Oxford University.