Its creators aim to deliver insights into why some people get the disease more severely than others.
They also hope to create a map showing where outbreaks are happening and help distinguish cases from those of the common cold.
It is one of many such new apps. Experts have warned people to be cautious about which they download.
At present, Covid Symptom Tracker is the third most popular app in Apple's UK store and second in Google Play's new releases chart for the country.
Its developers are targeting one million downloads in 24 hours.
The program has been shared thousands of times via WhatsApp and other social-media platforms.
Created in just three days by researchers at Guy's and St Thomas' hospitals and King's College London university, in conjunction with the nutrition advice start-up Zoe, it has already reached 750,000 downloads and, according to its developers, is being installed at a rate of 50,000 times an hour.
Home-testing kit
The app was the brainchild of Prof Tim Spector, a genetic epidemiologist at King's College, who has specialised in the genetics and medical histories of twins for the past 25 years.
"I was rather depressed as they were shutting down everything in the university and I thought that twins are the best studied people in the country, so how can we use that information in this crisis?" he told BBC News.
Initially, the app was made available only to the twins taking part in his studies, who were sent a home-testing kit to better understand which symptoms corresponded to the coronavirus.
But the professor realised it could be scaled up to the general public, without the testing element.
Temperature reading
"The NHS hasn't come up with a better alternative and this app seems to be working," he said.
"We are hoping to get to one million downloads by the end of the day and we will also be ready to release data by then for the NHS, data modellers and researchers to play with."
The software requires users to share personal details, including their age, height and medical history.
It then asks them to describe symptoms, if they have any, on a daily basis, as well as to give a temperature reading.
Critically unwell
Prof Spector said it could potentially help the NHS:
- learn how fast the virus is spreading in a particular area, as well as highlighting high-risk parts of the country
- better understand the symptoms, including the differences between those of the virus and the common cold
- explain why some people develop a mild illness while others become critically unwell
A spokeswoman for Zoe told BBC News all shared data would be anonymised and not used for any commercial purpose. And users could delete all their records when the crisis was over.
But as spam and malware skyrocket on the back of the pandemic, one expert said people would be wise to be cautious about downloading other apps purporting to help tackle coronavirus.
"I am concerned by the rash of websites and apps intended to allow people to report of their Covid-19 symptoms," said privacy expert Pat Walshe.
'Dubious ethics'
"I've found it difficult or impossible to determine who is behind a number of them.
"They do not adopt appropriate standards of compliance with data protection law and I see dubious ethics.
"Could an app help? Yes, possibly. But I think we need the NHS to coordinate it in order to provide confidence, trust and protection."
Prof Spector agrees people need to be careful.
"There are lots of scams out there and bogus things trying to get your details," he said.
People wishing to download the Covid-19 tracker can do so from Apple and Google's app stores.
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