In a statement, the feared FSB said it had detected malware "intended for use to spy online on some 20 institutions in Russia".
The organisations in question were government, scientific and military institutions, the FSB said.
"The circumstances appear to indicate... a planned attack, designed by professionals," it added.
There was no indication who the Russian intelligence service believed might be behind such a plan.
According to the FSB, the virus was sent as an attachment in an email, allowing the sender to intercept data traffic, listen in on phone calls, take screenshots, switch on microphones and cameras and log keystrokes.
The statement came as US Democrats said Saturday they had been targeted by yet another cyber-attack.
A hack on Democratic National Committee servers resulted in last week's embarrassing leak of emails that revealed how party leaders sought to undermine Clinton's Democratic White House rival Bernie Sanders.
Russia has denied intervening in the US election campaign, though Republican candidate Donald Trump sensationally challenged Moscow to hack into his rival Hillary Clinton's emails.
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