The six arrested are accused of "public indecency" and "inciting immorality among young people", the sources said following Monday's arrests.
Egypt's state security prosecutor has been tasked with carrying out the investigation into the incident on Friday at a concert of Lebanese band Mashrou' Leila, according to a judicial source.
Egyptian law does not expressly ban homosexuality, but gay men have previously been arrested on charges including debauchery and contempt for religion in the deeply conservative Muslim society.
In April 2016, an Egyptian court sentenced 11 men to jail terms of up to 12 years after convicting them of "debauchery and incitement to debauchery", provoking international criticism.
Egyptian police use US-based social networking application Grindr to monitor homosexuals.
Monday's rainbow flag incident has sparked controversy pitting conservatives against advocates of civil liberties in the media and on social networks.
Egypt's conservative musicians syndicate said Mashrou' Leila, whose singer Hamed Sinno is openly homosexual, would not be authorised to perform again in Egypt.
The group has previously been banned from performing in the Jordanian capital.
