Since September 11 dozens of schools, universities, malls, railway stations, airports and government offices have been evacuated and inspected by law enforcement in big cities all over Russian Federation - from Kaliningrad in the West to Vladivostok in the Far East.
Neither the emergency services in Moscow nor the police could immediately confirm the incidents to AFP.
Three of Moscow's main train stations were being checked on Wednesday, along with the city's famous GUM department store near Red Square, and more than 20 other buildings, the news agency said.
"Apparently, this is nothing else but phone terrorism, but all these messages should be checked", the source said.
He added: "The only goal is to set off destructive processes; to sow panic". Also, students and teaching staff were evacuated from the Sechenov Medical University and MGIMO International Relations University.
"It is possible this could be preparatory work for a serious terrorist attack".
An emergency services source told news agency Tass: "Evacuation has been conducted at 30 facilities, with a total of 20,000 people taken outside".
Many were city halls, schools and other official buildings, Bloomberg reported, citing local news reports.
According to reports, those responsible for the threats were still not located.
The Interior Ministry and Security Council declined to comment.
Police were also investigating threats at several train stations. Initial reports indicated the evacuations might have been some kind of exercise.
Russian Federation has been hit by militant attacks, including a bombing on the St Petersburg metro in April in which 16 people died.