Car maker Rolls-Royce said it was "approaching a critical point" due to workers having to self-isolate and may have to halve production if the trend continues.
The company, owned by Germany’s BMW, said it was "extremely concerned" at the number of staff at its manufacturing site in Goodwood, West Sussex, being ordered to stay at home because of NHS Test and Trace app alerts.
Meanwhile, Nissan’s UK plant has been affected by the same issue with more than 700 workers at the site in Sunderland reportedly self-isolating.
Ministers have been urged to act after trade union Unite said that across the country, factories were on the verge of shutting down due to the worker shortages.
Rolls-Royce Motor Cars said in a statement that it was currently running at full production.
It said: "We operate a two-shift system and every car is hand-built to customer order.
"We are extremely concerned at the number of employees at our manufacturing plant who have been forced to self-isolate after being notified by the NHS Test and Trace app.
"We are approaching a critical point – it will be necessary to reduce our two shifts to one if numbers of self-isolations continue to rise, effectively halving production.
"This would be deeply detrimental to our customers’ expectations and our business."
At Nissan’s Sunderland factory, more than 700 workers were reportedly forced to stay off work and isolate, leading to a number of shifts being cancelled entirely.
A spokesman for the car maker said: "Production in certain areas of the plant has been adjusted as we manage a number of staff being required to self-isolate following close contact with COVID-19.
"The wellbeing of our team is our number one priority and we remain confident in the rigorous safety controls we have on site."
Stephen Phipson, chief executive of manufacturers’ association Make UK, told Sky News that over the past 10 days there had been a "very significant rise" in the number of workers in the sector being required to stay at home.
Mr Phipson said that in many companies it had resulted in one in five being off.
He added: "In a factory setting, you can’t work from home – this directly affects the output."
Steve Turner, Unite assistant general secretary, said: "No one is advocating for coronavirus controls to go out the window and Unite’s number one priority remains the health and safety of our members.
"But the reports Unite is receiving from our members and their employers are extremely worrying.
"It is not an exaggeration to say factories are on the verge of shutting and that at some sites hundreds of staff are off work."
Mr Turner warned that people would simply start deleting the app if a solution was not found soon.
For the car industry, it is the latest headache with a shortage of semiconductor chips already holding back production.
Mike Hawes, chief executive of the Society of Motor Manufacturers and Traders, said: "Like many other sectors of the UK economy, automotive manufacturers and suppliers are being impacted by the growing number of coronavirus cases as restrictions are eased.
"The industry has taken every step to make premises COVID safe, but with transmissions rising outside the workplace, and self-isolation rules preventing staff from working, staff shortages are putting production at risk and undermining the sector’s recovery.
"Urgent action is needed to mitigate this impact, such as a change to the sensitivity of the NHS Test and Trace app or bringing forward the 16 August target date for exempting fully vaccinated adults from self-isolation."
Car makers Vauxhall and Jaguar Land Rover, however, told Sky News they were not experiencing problems.
JLR is part of a government pilot scheme which means workers who receive a COVID alert can take daily lateral flow tests rather than having to self-isolate.
Outside the car sector, aircraft engine maker Rolls-Royce – which is separate from the car maker of the same name – said that like other employers it has had to adapt when workers are told to self-isolate.
"However, more than 9,000 production employees in the UK have been working at our sites for the majority of lockdown to ensure the ongoing operation of key manufacturing facilities," the group said.
"During this time, we have introduced new ways of working including split shifts, rotas, additional cleaning and more physical dividers to protect our employees, and will continue to do so for the foreseeable future."
Earlier this week, passengers faced long queues at Heathrow after security staff were told to self-isolate.
Images posted online show lines of travellers inside Terminal 5, with some describing it as "total chaos".
Hospitality and retail industry leaders have also said this week that they are facing big staff shortages due to workers receiving alerts from the Test and Trace app.