Last month, world record-holder Yuliya Efimova was banned for 16 months after testing positive for a banned steroid. Two other Russians are serving suspensions after failing drug tests and at least five were banned last year, mostly for domestic violations.
The cases come as Russia prepares to host next year's world swimming championships in Kazan.
"The number of breaches of anti-doping rules in the international arena is on the brink," Mutko told Russian media. "One or two more breaches and the federation could be suspended altogether."
Mutko said the Russian Swimming Federation is one of "several" Russian sports bodies at risk of suspension over doping scandals. He did not name any others.
According to the statutes of swimming's world governing body FINA, a member federation receives a two-year ban if four of its competitors in the same discipline commit doping offences within 12 months, with certain exemptions.
FINA executive director Cornel Marculescu told The Associated Press that Russia is not currently in that category.
"I will not further comment on this possibility," he said.
Marculescu said that he and FINA remained "fully confident" in Russia's ability to host the world championships.
"The facilities are amazing and FINA is receiving a great support from the authorities of the Russian Federation and of the Republic of Tatarstan," he said, referring to the Russian region where Kazan is located.
"The personal commitment of Mr. Mutko, Russian Minister of Sport, and of Mr. (Rustam) Minnikhanov, President of Tatarstan, are strong guarantees that FINA's major event will be a success."
The world championships will be held at venues from last year's World University Games, including the Kazan Arena, which will also serve as one of the stadiums for the 2018 World Cup.
Efimova will be eligible to complete in Kazan because her ban for testing positive for the steroid DHEA expires next Feb. 28.
Two Russian swimmers are currently awaiting verdicts in their own doping cases in international competition.
Sergei Makov tested positive for the banned muscle-building substance ostarine at a World Cup event in Moscow in October. Vitalii Melnikov tested positive for the blood-boosting hormone EPO at last year's European short course championships in December.