According to the text of the bill published yesterday by the Ministry of Education, school and university officials should "analyse the personal sites of students and personnel" and compile reports on those "who have a tendency towards breaking the rules".
The bill also says it is their responsibility to "continuously monitor the Internet" in order to collect data on "terrorist activities" in their region.
All information on any unusual behaviour of a student must be sent to the FSB security service, the successor to the KGB, it says.
"This amounts to a bureaucratic absurdity, or more likely a tentative attempt to put students and teachers who do not approve of the current government under surveillance," Andrei Demidov, head of a teachers union, told the Gazeta newspaper's website.
The Ministry of Education responded that the bill was only at the consultation stage and was not finalised, reported Kommersant daily.