Raymond Lidal, 28, was detained in late March when he failed to show a journalist's visa while filming Saudi-led airstrikes against Huthi Shiite rebels who have seized the city, according to the Verdens Gang newspaper (VG).
"Verdens gang has learned from several sources... That the Norwegian was suspected of espionage and was accused of carrying out some kind of investigation," the paper said.
Norway's foreign affairs ministry confirmed that a Norwegian national had been arrested in Sanaa but would not comment on his identity or the reasons for the arrest.
The ministry added that details of the case were unclear due to the "chaotic situation" in the country.
Norwegian public broadcaster NRK - one of Lidal's employers - said he had been reporting from Yemen for several media outlets since 2013 while studying there.
Iran-backed Huthi rebels have seized swathes of territory in Yemen since they entered Sanaa in September forcing the government to flee and prompting airstrikes since March 26 by a Saudi-led coalition of Arab states.