Majid al-Mohandis was performing at a festival in the western city of Taif when the woman darted on to the stage.
Videos posted online showed her holding on to Mr Mohandis while security staff tried to pull her off him.
Women in Saudi Arabia are not allowed to mix in public with men who are unrelated to them.
Mr Mohandis, whose website says is “the prince of Arab singing”, has not commented on the incident. The Iraqi-born singer, who also has Saudi citizenship, continued to perform after the incident.
A public prosecutor will now consider harassment charges against the woman, police told Okaz, a leading Saudi newspaper, and Efe news agency.
The country has strict morality laws regarding alcohol prohibition, modest clothing and gender segregation.
Restrictions that had long been placed on women attending public events in the kingdom have been relaxed in the past year under a series of reforms by Crown Prince Mohammed bin Salman.
Vision 2030, unveiled last year, aims to increase household spending on cultural and entertainment activities in the oil-dependent kingdom from 2.9% to 6% by 2030.
Women were allowed to attend a concert and a football match for the first time on the same weekend, and the kingdom also hosted its first concert by a female singer, Lebanese star Hiba Tawaji, in December. Women were also allowed to drive from last month.
But significant restrictions remain in place and women still have to adhere to strict dress codes. The unnamed women seen hugging Mr Mohandis was wearing a niqab, a headdress that shows only the eyes.