But Mutko, a member of FIFA's executive committee, added that drastic change was needed at "a world class organisation" that "needed to reclaim the trust of fans".
Speaking to a group of foreign journalists assembling in Moscow at the start of a tour of the 11 World Cup cities, Mutko said: "No change in management at FIFA will change anything."
Turning to the 2018 finals his country is hosting he said: "The World Cup does not belong to Russia, it belongs to FIFA but Russia will organise the World Cup for FIFA as planned.
"There is no threat to us here. The World Cup is a gem which has to be valued and it will be."
The FBI is investigating widespread allegations of FIFA corruption going back more than two decades, while the Swiss authorities are probing the voting mechanism that saw Russia win the right to stage the next edition and Qatar the 2022 finals.
But Mutko, who is also the CEO of the 2018 Organising Committee and has repeatedly said his country's bid was clean, brushed aside the investigations as far as Russia was concerned.
"As the sports minister of Russia, I could suggest many other federations where the FBI might want to investigate," he said without elaborating. "There are 209 countries in FIFA."
STAUNCH ALLY
Russia has been a staunch ally of outgoing FIFA chief Sepp Blatter over the years and have supported him during his 17 years as president, and Mutko praised the 79-year-old Swiss.
Blatter said last month he was "laying down his mandate" just days after winning a fifth term and less than a week after police arrested seven FIFA officials in a dawn raid in Zurich.
"Mr Blatter has worked for FIFA for 40 years and is a world class official who, during that time, has made FIFA into a world class organisation and one of the most powerful organisations in the world," said Mutko.
"But the image and the profile now needs a drastic change to stabilise the situation and bring it back to what it was.
"Our football family needs to get over what has happened and needs to regain the trust of fans around the world, and I think it will be able to regain that trust as long as no-one's goal is to kill off football."
FIFA's executive committee meets on July 20 in Zurich to determine when an extraordinary congress will be held to elect a new president on a date between December and March next year.
(Reporting by Mike Collett; editing by Ken Ferris)
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