News site Nikkei, which first reported the shutdown, said supplier Kojima Industries Corporation suspected it had been hit by a cyber-attack, causing a halt in production.
Japanese factories account for about a third of Toyota's production.
It told the Wall Street Journal it did not know whether the factories would remain closed beyond Tuesday.
Toyota is the world's best selling carmaker.
Its production target for this year is 8.5 million vehicles.
And the closure will reportedly set this back by about 13,000 cars.
"Due to a system failure at a supplier in Japan, we have decided to suspend the operation of 28 lines at all 14 domestic plants", Toyota told news agency AFP.
Like other manufacturers, Toyota's production has been hit by the global semi-conductor shortage.
The shutdown included some plants operated by Toyota-affiliated Hino Motors and Daihatsu Motor, Reuters reported.