Russia was banned from international competition after a damning report by the World Anti-Doping Agency (Wada).
But Vitaly Mutko argued not lifting the ban for the 2016 Rio Olympics would be "unfair and disproportionate" and that clean athletes should not be punished.
They will have been rigorously tested by Rio, he wrote in the Sunday Times.
Wada's report depicted a culture of systematic state-sponsored cheating by Russia's athletes, with even the secret services involved.
It said the All-Russia Athletics Federation (Araf), the Russian anti-doping agency (Rusada), and the Russian Athletics Federation had failed to comply with anti-doping procedures.
"We do not deny having a problem in Russia, and we are doing everything possible at state level to eradicate doping, including punishing athletes and coaches found to have violated anti-doping rules," Mutko wrote.