The Transport for London (TfL) trial follows the success of its "Baby on board" badge for pregnant women.
TfL is recruiting 1,000 people to start wearing the blue badges from 12 September.
Mayor of London Sadiq Khan said he hoped they would "give confidence" to people who find standing difficult.
Some travellers, such as James McNaught who is joining the trial, had already started making their own badges to alert fellow passengers to their condition.
The 45-year-old designed "Cancer on board" badges after travelling on the Tube between Kentish Town and University College Hospital for chemotherapy.
Radiotherapy on his throat left him unable to speak to ask for a seat, and the morphine made him appear drunk.
"I'm really pleased TfL is doing this trial," he said. "A badge and card could help make a real difference to the lives of people undergoing drug treatment or with longer term conditions or disabilities."
TfL will use social media and customer information to encourage other passengers to look out for the badges.
"This small act of consideration from Londoners could make a huge difference to disabled people getting around the city and being fully involved in all London has to offer," said Alice Mitchell-Pye of charity Leonard Cheshire Disability.
The six-week trial is believed to be the first of its kind in Europe.
People with hidden health conditions are being offered "Please offer me a seat" badges in a bid to help ease their suffering on London transport.