Two of the men were among the kingdom’s most influential figures.
The detentions have been linked to Crown Prince Mohammed bin Salman.
In 2017 dozens of Saudi royal figures, ministers and businessmen were confined to the Ritz-Carlton hotel in Riyadh after the crown prince ordered their arrests.
Mohammed bin Salman is considered the de facto ruler of the kingdom, after he was named crown prince by his father in 2016.
The detentions took place early on Friday morning, report the New York Times and Wall Street Journal.
The three men reported to have been arrested are the king’s younger brother Prince Ahmed bin Abdulaziz, former crown prince Mohammed bin Nayef, and a royal cousin, Prince Nawaf bin Nayef.
Mohammed bin Nayef was interior minister until he was removed from his role and placed under house arrest by Mohammed Bin Salman in 2017.
Guards arrived at the homes of the royals wearing masks and dressed in black, and searched their homes, the Wall Street Journal says.
Mohammed bin Salman won international praise in 2016 when he promised a series of economic and social reforms to the deeply-conservative country.
However he has been embroiled in a series of scandals including the murder of Saudi journalist Jamal Khashoggi at the Saudi embassy in Istanbul in 2018.
In recent days Saudi Arabia has been taking measures to contain the spread of the deadly new coronavirus.
Foreign pilgrims have been prevented from entering the country to perform Umrah, or pilgrimage, and there are questions over whether the hugely important annual Hajj will go ahead this year,
On Thursday Islam’s holiest city Mecca was emptied to allow a deep clean.