"We can see that there is bigger interest from foreign intelligence services to penetrate and find secret information in our country," Anders Thornberg, head of Säpo, told Radio Sweden in an interview after the report was released.
He refused to give details on who had been behind last year's intrusion attempts.
"Every year and every month we see attempts to recruit people and find information in unlawful ways. And there must be a dark figure also of course, but we cannot say."
But he said that the greatest threat came from nation states.
"Generally it is Russia, China and Iran at the time being, but it can come from wherever: it depends on the situation. So we are looking in all directions, but for a very long time the biggest threat has come from Russia."
Säpo section chief Fredrik Agemark gave a presentation outlining the threat at a press conference on Thursday.
"Under 2016, the security police worked on several cases where people in agencies crucial to security were aiding foreign powers," he said.
Several suspected spies had applied for jobs within Swedish agencies, and foreign powers had also tried to recruit people who were already working at Swedish agencies.
Swedish public authorities had also faced cyber-attacks aimed at securing information.
"It's not always easy to see who is behind the attacks, but one can see that it is a well-resourced entity," Agemark said.
As well as planting agents, foreign powers have also exploited weak cybersecurity and security failings in government buildings to access secret information.
"The security police have never had a bigger or more complicated mission," Thornberg said at the press conference, blaming the increasing workload on Sweden's "growing military strategic significance".
None of the people investigated have so far been charged with espionage.
"Säpo waits until someone becomes a full-blown spy before we make any arrests," Thornberg said.
Home Affairs Minister Anders Ygeman said the report underlined the urgency with which Sweden's agencies needed to tighten up their security.
"Some public authorities have not yet gone all the way," said Ygeman. "That's why we need to strengthen the security services in their work to help the authorities take the necessary first step, which is a good security analysis."