Three nuclear sites have been targeted and sightings have been reported over airports and the royal palace.
Authorities have not speculated on who is behind the mysterious drones.
Police and the coastguard are searching the sea and islands around Stockholm, local media reports say.
The latest sightings on Monday evening involved a drone above the Forsmark nuclear plant, but security agency Sapo said it was also investigating earlier drone flights near the Ringhals and Oskarshamn power plants. Police appealed to the public to come forward with information.
Sapo said the drones were suspected of “grave unauthorised dealing with secret information”.
“The security service is conducting the investigation regarding the drone flights over our nuclear plants. Regarding other drone observations it’s so far a matter for the police,” Sapo official Fredrik Hultgren told the BBC.
He refused to give details about the type of drone, but they have been widely described as military-style and as having large wings.
Swedish TV reported that a police helicopter was following a drone flying above it at a height of up to 1,000m (3,280ft) to the south-west and north-west of the capital Stockholm.
According to the Aftonbladet news site, a drone was seen over the weekend circling the parliament and government buildings, as well as the royal palace in Stockholm. Drones were earlier spotted near Kiruna and Lulea airports.
Prosecutors are keeping an open mind over who is behind the drone flights. They come at a time of increased military alert in Sweden because of tensions between Russia and the West over the Russian military build-up near the borders of Ukraine.
Sweden has recently deployed troops to the Baltic island of Gotland in what officials describe as a signal that Sweden’s military forces are ready to defend their territory.
Sweden is not a member of the Nato defence alliance but said it was responding to three Russian landing craft moving into the Baltic Sea through Denmark’s Great Belt Strait.