The group had staged an illegal protest outside the presidential residence in the capital, Kigali, police said.
Protests outside Mr Kagame's home are extremely rare, correspondents say.
The opposition accuses him of running an authoritarian regime since he took power in 1994.
However, his supporters and Western governments credit him with ending a genocide that killed some 800,000 people before his rebel forces seized power in Rwanda.
The BBC's Jean Claude Mwambutse reports from Kigali that those who were arrested came from the little known Intwarane group, which is not recognised by  the official Roman Catholic church.
Police chief Emmanuel Gasana said the group was arrested as it was heading towards Mr Kagame's home in Kigali's wealthy suburb of Kiyovu.
Eleven members of a breakaway Catholic group have been arrested in Rwanda after they tried to deliver a "vision from Virgin Mary" to President Paul Kagame, urging him to make reforms.