François Hollande enjoyed a rare moment of support and sympathy from across the political spectrum on Friday after a celebrity magazine claimed he was having an affair with an actress.

Whether it was true did not matter. The blatant breach of France's draconian privacy laws prompted outrage and a show of unity that has until now eluded the French president, whose popularity has plummeted since he took office in 2012.

Even Marine Le Pen, leader of the far-right Front National, who has never let a good word about Hollande cross her lips, was forced to admit she was shocked by Closer magazine. "Everyone has the right to have their private life respected," she said on Friday, adding "As long as it doesn't cost the taxpayer a centime."

Presidential affairs are the rule rather than the exception in France, but a largely compliant media and a tacit omerta have often ensured they do not become public knowledge.

Closer's special edition and seven-page article, with photographs, appear to have taken France's political class by surprise. The magazine claimed the president had been crossing Paris on the back of a scooter to make clandestine visits to Julie Gayet, 41. Photos apparently taken the following morning show Hollande's bodyguard arriving at the her flat with what Closer said was a bag of croissants.

In a statement from the Elysée Palace, Holland, 59, did not directly deny the report. Instead he accused the magazine of invading his privacy and said he was considering legal action against Closer, which caused a storm in 2012 after publishing paparazzi photographs of the topless Duchess of Cambridge on a private holiday in the south of France. The statement said Hollande "deeply deplores the attacks on the principle of respect for privacy, to which he, like every citizen, has a right".

There was no statement from Hollande's partner, the former Paris Match journalist Valérie Trierweiler, 48, who is considered to be France's first lady. He has four children with the politician Ségolène Royal, who ran against Nicolas Sarkozy in the 2007 presidential race.

Rumours about the president and the actress are nothing new. Last March Gayet began legal action to identify who was behind internet rumours that she was romantically involved with the president. During Hollande's 2012 election campaign she took part in a political broadcast in which she raved about the Socialist candidate describing him as "humble", "incredible" and a man who "really listens".

On Friday afternoon, Laurence Pieau, editor of Closer, promised to remove the story from the magazine website, following a demand from Gayet's lawyer. It was not clear if any legal proceedings had been formally started, but Pieau said her magazine was under a "very clear injunction".

"Julie Gayet's lawyer contacted us to ask that we remove from our website all mention of this relationship and to do the same with Google," Pieau said, adding that she had not been asked to withdraw the magazine in which the photographs appeared. "We will withdraw the news , probably this evening [Friday] because it is a very clear injunction. However, we have had no contact with the Elysée." Pieau said there were no plans to withdraw the print edition. She added: "When Closer published the first photographs of François Hollande with Valérie Trierweiler in 2012, François Hollande had sought the removal of the magazine from the news stands but this was not granted. This time, the Elysée has not asked."

She said such demands were rarely successful except in cases of a threat to public order.

Earlier in the day, Pieau had dismissed the uproar over her magazine's report. "He's a normal president, a normal person. He's a president who's fallen in love and an affair. Everyone has to calm down over the photographs," she said.

The scandal highlighted the gulf between attitudes to privacy either side of the Channel. Professor Matthew Fraser, a media specialist at the American University of Paris, said: "French law is indifferent to the 'truth', unlike in the UK and America where the press is free to print something if it can be proved to be true. In France, it's illegal to invade personal privacy even if it's the truth, so it's more than a cultural difference, it's the law."

However, social media meant it was difficult for politicians and celebrities to hide behind French privacy laws, he added. "It's challenging, to put it mildly, for public figures to separate public and private in the age of social media. In the past, public personas were stage-managed through established media dominated by professional journalists. The public knew very little about the private identities of politicians. Only journalists knew. But the old rules are out. With social media, information is accessible and goes viral in a matter of minutes. You can't manage or control it."

In response to suggestions the scandal could actually improve Hollande's popularity with electors, Fraser was dubious. "In contrast to the conquests of Hollande's philandering predecessors, this affair seems almost comical – like a Feydeau farce – an impression that is made more vivid by the image of him in a motorcycle helmet scootering around Paris for secret assignations."