The threat is hanging over Russia after the World Anti-Doping Agency (WADA) revealed they would push for such a sanction if its latest Independent Commission found evidence of state-supported doping at the 2014 Winter Olympics in Sochi.
The results of the report are due to be revealed no later than July 15.
The Commission, chaired by Canadian lawyer Richard McLaren, was launched after Grigory Rodchenkov, the former head of the Moscow Anti-Doping Laboratory, claimed that up to 15 Russian medal winners at Sochi 2014 were implicated in a clandestine operation in which doping samples were switched for clean ones.
Rodchenkov, the subject of a criminal case from Russian Federal investigators, alleged that the state-sponsored scheme involved athletes ingesting a "three-drug cocktail" of banned steroids which were mixed with alcohol.
It also featured as a covert system to replace the urine of affected medal winners with clean samples using soda containers and baby bottles, he claimed.
"If all the Russian national team gets suspended, it would be a great failure for me," Mutko told the Russian Match TV broadcaster.
"I am ready to take all the responsibility and resign."
The International Association of Athletics Federations (IAAF) last week decided to extend a ban on the All-Russia Athletic Federation (ARAF) imposed last November following the publication of the first WADA Independent Commission which found evidence of state-supported doping.
Russia's weightlifters are also facing the threat of being banned from Rio 2016.
They are one of three countries, along with Belarus and Kazakhstan, who w ill be suspended for one year by the International Weightlifting Federation in the event that it is proven the countries produced three or more anti-doping rule violations in the combined re-analysis of samples from 2008 and 2012 Olympics in Beijing and London respectively.
The 57-year-old Mutko has held the portfolio for sport in the Russian Government since May 2008.
He is also President of the Russian Football Union and a member of FIFA's ruling Council, giving him an important role in his country's preparations to host the 2018 World Cup.
Mutko claimed the IAAF should be disbanded following the decision of its ruling Council in Vienna last week but the stance of the world governing body has been backed by WADA.
The ARAF are currently in the process of launching an appeal to the Court of Arbitration for Sport in Lausanne against the IAAF.
The appeal is expected to be submitted in tandem with several from individual athletes, including double Olympic pole vault champion and world record holder Yelena Isinbayeva.
Mutko revealed that, at the same time, Russian athletes will be encouraged to apply to the IAAF to compete as neutral athletes.
An amendment to Rule 22.1A was passed by the IAAF means that if there are any individual athletes who can clearly and convincingly show that they are not tainted by the Russian system because they have been outside the country and subject to other effective anti-doping systems, then they should be able to apply for permission to compete in international competitions, not for Russia but as a neutral athlete.
"Up to 67 athletes will submit individual applications to take part in the Olympic Games and simultaneously will appeal to the CAS to prove their cleanness," Mutko told Match TV.