Russia are sending up to 450 competitors to London and Mutko expects a strong showing in boxing, gymnastics, synchronised swimming and athletics where world record-breaking pole vaulter Yelena Isinbayeva will try to retain the title she won in 2008 and 2004.
"We will not catch up with the United States or China at this Olympics but we can compete with Britain and Germany for third place in the medals table," he said.
"It will be very difficult to win more medals than Britain who are at home ... but if we end up fourth this will also be no tragedy."
The Olympics may mark the first visit by Russian President Vladimir Putin to Britain in nine years, his spokesman said earlier in June amid tense relations between London and Moscow over security, diplomacy and human rights issues.
Russia will host the 2014 Winter Olympics at the Black Sea resort of Sochi and soccer's World Cup in 2018.