The government also wants people to provide short voice and breathing sound samples as it looks at developing an app for wider use.
US researchers have already created an algorithm that detects Covid from coughs with a near-100% accuracy rate.
It is hoped this type of technology will make diagnosis cheaper and easier.
For the Department of Health and Social Care's research project, adults who have taken a Covid test in the last 72 hours can submit the barcode they have received from the NHS via their mobile, laptop, tablet or desktop computer.
They then record their breathing, coughing and a read-out sentence.
Researchers compare the test results - showing an infection or a clean bill of health - with the recordings, as they seek to refine the technology.
A team at the Massachusetts Institute of Technology developed an algorithm last year which picked up 98.5% of Covid infections.
The accuracy rate was 100% among participants who had reported no other symptoms.
Last year, the UK awarded two contracts to Japanese technology conglomerate Fujitsu to explore potential applications of voice and cough data.
It also recently awarded a contract to Ipsos Mori for data collection.