The Federal Security Service, or FSB, said the suspects were preparing to launch the attacks on Friday, the beginning of the school year in Russia.
The agency said one of the suspects was planning to carry out a suicide bombing. It said a powerful homemade explosive device was found at his apartment.
Another man was preparing to attack civilians with an ax and put out a video statement pledging adherence to the IS.
Earlier this month, the FSB arrested four suspects accused of planning suicide bombings on Moscow's transit system and shopping malls.
In May, the FSB arrested another group of suspected IS members who were also accused of plotting terror attacks in the Russian capital.
The arrests follow a suicide bombing in the St Petersburg's subway that left 16 dead and wounded more than 50 in April.
President Vladimir Putin said in April that some 9,000 militants, about half of them from Russia and the rest from ex-Soviet Central Asian nations, have joined the Islamic State in Syria.
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