Russia was suspended by governing body the IAAF for state-sponsored doping in November 2015, a ban extended on Monday to cover the 2017 World Championships.
Mutko, who was criticised by the IAAF on Monday, accepted "there were many abuses and breaches" but says "colossal work has been done" over the last year.
Russia expects to be back by November.
A return to competition will require the reinstatement of the national drug-testing agency, which remains suspended over various allegations of covering up doping.
Mutko - who assumed his role in October - has been outspoken since the McLaren report alleged there had been a state-backed Russian doping programme. In January he said female doping results can be distorted if athletes have had sex in the days leading up to a test.
But on Monday he said: "Athletes broke the rules and many coaches don't understand how to work without doping and it's high time for them to retire."
Russia will miss the World Championships which begin on 4 August in London, but some athletes from the country could compete in London under a neutral banner.
The IAAF is so far considering 35 applications from athletes, who could be included if they can show a record of independent drug-testing by agencies other than the suspended national body.
Russian track federation vice president Andrei Silnov said he was against the inclusion of neutral athletes in principle but also questioned if the country's problems were as bad as the IAAF outlined.
"It's all being solved, slowly but surely. We're doing what we need to do," Silnov said. `"They say we have a culture of doping. What culture of doping?"
Russian coaches who "do not understand how to work without doping" should "retire" says the country's deputy Prime Minister Vitaly Mutko.