"The participation of Russian athletes at the Olympic Games in Rio greatly depends on the results of the WADA investigation," Bach said.
"Should there be evidence of an organized and comprehensive system of doping, which affects several sports, the International Federations and the IOC must make the difficult decision to choose between collective responsibility and individual justice."
On June 17 the International Association of Athletics Federations (IAAF) is set to meet in Vienna and decide whether the provisional suspension of the Russian Athletics Federation that it imposed in November 2015 will be lifted, or stay in place and prevent Russian athletes from competing in the Rio Olympics.
On Tuesday the IOC revealed that retesting of samples taken from athletes at the 2008 Beijing Olympics, using more up-to-date techniques, has discovered evidence of illegal substances used by 31 unidentified athletes from 12 countries, who now face bans from competing in Rio. It has not revealed which countries the athletes are from.
"Sportspeople who are doping, regardless of what country they come from, must be punished and stopped from competing. We support and are ready to offer complete assistance to WADA investigations, and we have told the agency this," said spokeswoman Natalya Zhelanova.
"But the responsibility is for individuals. Clean sportspeople who have devoted years of their life to sport, trained in good faith and observed all the rules and regulations, including anti-doping, should not be deprived of the right to take part in competition."
The newspaper also reported that the US Justice Department has begun its own investigation into the allegations of doping in Russian sport.
"It is surprising that the US justice system has decided to investigate Russia in particular. You can look at WADA's figures and see that there are at least 1,000 cases of doping in world sport, they could investigate all of them. I would like to recommend that the US investigate its own team," responded Russian Sports Minister Vitaliy Mutko.