"The governments of cantons Basel Countryside and Basel City call on the French authorities and (French energy giant) EDF to switch off and immediately shut down the Fessenheim nuclear power plant until it undergoes an intensive check that takes into account the events in Japan," they said in a joint statement.
The cantons said they had in past years raised questions about the risks surrounding the power plant located 35 kilometres (20 miles) north of Basel.
"The current, serious nuclear accident in Japan reinforces our opinion that the Fessenheim nuclear power plant poses a serious threat for the populations in the region," they added.
Built in 1977-78, Fessenheim is France's oldest nuclear plant that is still in operation, said the cantons, pointing out that the reactor also lies in a quake-prone area.
The cantons said they planned to lodge their request with Alsace's regional government on April 1.
The crisis surrounding Japan's Fukushima No.1 power plant, which was damaged during a powerful earthquake, has prompted several governments to monitor their nuclear reactors.
Following the crisis, the government suspended plans to replace its ageing nuclear power plants amid fears of a nuclear disaster.