The Fukushima Daiichi plant -- around 140 miles from Tokyo -- was severely damaged by a March 2011 earthquake and tsunami in northeastern Japan.
The cartoons by the French satirical newspaper Le Canard Enchaine follow a similar development last year involving a French TV program, Kyodo news agency reported.
On Saturday Tokyo won the bid to host the 2020 Olympics and on Wednesday Le Canard ran an article with two offending cartoons.
One cartoon shows a sumo wrestling contest near a crippled nuclear power plant. A wrestler with three arms is fighting another with three legs, while a reporter is saying, "Marvelous! Thanks to Fukushima, sumo has been included among Olympic events."
Another cartoon showed a swimming pool with two people nearby in protective gear holding radiation testers. The caption was, "There is already a pool in Fukushima for the Olympics."
Kyodo news agency quoted Chief Cabinet Secretary Yoshihide Suga as saying the Le Canard report was "inappropriate" and "extremely regrettable."
He said a protest will be lodged directly with Le Canard through the Japanese Embassy in France.
In October, the French government apologized for a television program by public broadcaster France 2. The program showed a composite picture of Eiji Kawashima, goalkeeper of Japan's national soccer team, with four arms, citing the "effect" of the Fukushima disaster.
Kyodo reported Suga saying such cartoons could "hurt the feelings of those affected by the disasters and give misunderstandings about the toxic water problem."
Prime Minister Shinzo Abe reassured the International Olympic Committee in Buenos Aires his government had mitigated the effects of a massive buildup of radioactive water at the plant and there was no threat to the general population.
"Let me assure you the situation is under control," Abe said. "[The Fukushima plant] never has done, and never will do, any damage to Tokyo."
Japan's government remains sensitive to criticism of its handling of the disaster. It was the biggest nuclear accident since the 1986 meltdown of the Chernobyl power station in Ukraine, the International Atomic Energy Agency said.
The Huffington Post reported in July Tokyo Electric Power Co. had confirmed its Fukushima plant likely is leaking contaminated water into sea.
A report by Japan Times said contamination of the sea wasn't a major risk compared with other issues.
"Compared with the release of radioactive material in the initial stage, the amount of material now is overwhelmingly small," Jota Kanda, a professor at Tokyo University of Marine Science and Technology, told Japan Times.
Around 300 tons of toxic water flows into the Pacific each day.
"This isn't something that has a big impact on fish in the sea," Kanda said.
The bigger threat is the highly radioactive water sitting in the more than 1,000 huge tanks on land, he said.
"I understand it's quite important to try to stop the groundwater [from entering the sea]. But I'm far more concerned about the tanks," he said.
"We still have extremely contaminated water in those tanks. In that sense, we are in a crisis-like situation."