The Russian foreign affairs department said in a statement that it had summoned France's ambassador Jean-Maurice Ripert to "give an explanation" of the incident.
"A plane of the French airforce flew dangerously close to a jet with a Russian parliamentary delegation headed by the Russian State Duma speaker Sergei Naryshkin," the statement said, adding the officials were travelling to a meeting of the Inter-Parliamentary Union in Geneva.
"The ambassador was made aware of Russia's deep concern over what happened," it said, adding that "actions like these by Paris undermine the possibility of using France as a place for multilateral meetings and talks."
France's foreign ministry rebuffed the Russian claim, saying that "it was a Swiss plane, an F18, and no French military plane is involved."
The Swiss defence department confirmed the incident, saying one of its F-18 warplanes had approached the Russian craft for "a routine check".
The department said the incident that occurred was "completely normal", reported the ATS news agency, which said that it occurred over the city of Bienne/Biel in the canton of Bern.
During the evening, Russian foreign ministry spokeswoman Maria Zakharova confirmed that Moscow had apologized to Paris through diplomatic channels, but had demanded an explanation from Switzerland.
"We have to work out what happened and why it was done," she said in televised comments.
Naryshkin's spokeswoman Yevgenia Chugunova told AFP that the warplane flew so close that members of the delegation could take a picture.
"I can confirm that this incident involving a French military plane happened this morning," she said.
"We saw it very close."
Naryshkin, who is on both the EU and US blacklists for publicly supporting the deployment of Russian forces to Ukraine and the annexation of Crimea, told Russian agencies he is aware of what happened although he did not see the warplane himself.
Another member of the delegation, Sergei Gavrilov, told TASS news agency that the approach took place "at the altitude of 3,700 metres above the Swiss border" and called it an "unfriendly act by Nato".
Last Wednesday, a Swiss air force F/A-18 plane crashed southeast of Besançon in the Doubs region of France in a training area used by French and Swiss air forces.
The 38-year-old pilot, who ejected prior to the crash, was injured and taken to hospital.
Swiss and French authorities have launched an investigation.