Health authorities in the southern region of Volgograd told state news agency RIA Novosti on Friday that 11 people had died from the virus and that nearly 50 schools had declared a quarantine.
One deadly swine flu case was also reported in Moscow, local authorities said today.
Six swine flu-related deaths have also been recorded in the southern region of Rostov and five deaths were reported in Saint Petersburg.
Swine flu deaths have also been reported in the Kostroma, Tambov and Kaliningrad regions, among others.
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The country's top doctor, Anna Popova, told Russian news agencies this week that "all necessary measures" were being taken to prevent the virus from spreading.
In post-Soviet Armenia, 18 people have died of swine flu since the start of the year, the South Caucasus country's health ministry said this week.
In nearby Iran, swine flu has left 112 dead and forced the hospitalisation of more than 1,000 since mid-November, authorities said late last month.
A major H1N1 outbreak sparked a World Health Organization pandemic alert in June 2009, after the virus emerged from Mexico and the United States.
The outbreak killed around 18,500 people in 214 countries. The alert was lifted in August 2010.