Hollande reacted furiously to the allegation -- backed up by photos reportedly showing the president entering the flat of actress Julie Gayet -- but did not deny it, threatening legal action over what he called an attack on his right to privacy.
Closer's Friday edition carried a seven-page report on the 59-year-old president's alleged infidelity under the headline "Francois Hollande and Julie Gayet - the president's secret love".
"It's a real passion that has... turned their lives upside down and makes them take insane risks," the magazine wrote.
To back up its claims, the magazine printed photos of Gayet, 41, arriving at a flat in an upmarket part of Paris on December 30, not far from the Elysee palace where Hollande lives.
Half-an-hour later, a man whom they identify as Hollande's bodyguard inspects the hallway of the apartment block before the arrival of a scooter with two helmeted men on board, one of whom they say is Hollande -- although his face is never revealed.
The alleged bodyguard, however, does show his face and is identified as part of Hollande's official security detail.
The next day, according to the photos, the alleged bodyguard arrives at the flat around 8am with a small bag that Closer says contains croissants.
Three hours later, a helmeted man is pictured mounting a chauffeur-driven scooter under the eyes of the alleged bodyguard.
Speaking in a personal capacity, Hollande slammed the report as an attack on the right to privacy, to which he "like every other citizen has a right".
The president, in a statement released to AFP, said he was "looking into possible action, including legal action," against the weekly magazine. But he did not deny the allegations.
France's political elite also backed the president.
“The President of France François Hollande, as a citizen, has asked for respect of his private life,” Prime Minister Jean-Marc Ayrault told French daily Le Parisien. “He is absolutely right and I have nothing to add to his statement.”
French Transportation Minister Frédéric Cuvillier also gave a full throated support of Hollande’s privacy when asked about the scandal in radio station RTL.
“Why would I want to a waste my time commenting on that type of information, if you can call it that,” he said. “I know that I’m the minister of transportation, but spare me this type of question.”
Rumours that Hollande is having an affair with Gayet, a mother-of-two, have swirled for months
In December, French actor Stephane Guillon made innuendos on the subject during a talk show where he and Gayet were invited to promote a film in which they co-star.
When the host asked Gayet about her public support for Hollande, Guillon - sitting next to her - started laughing. Asked why, he stuttered and finally said: "He would come on set. The president likes the film, his wife much less.
Hollande lives with his partner Valerie Trierweiler, a journalist for whom he left fellow Socialist politician Segolene Royal, the mother of his four children.
But the two are not married, and Trierweiler -- whom Hollande described as "the love of my life" in 2010 -- is referred to in the American media as "the first girlfriend".
If confirmed, though, the liaison will raise considerable concerns for the security of the president.
Respected news weekly L'Express last month reported that the presidential palace's security services were increasingly worried by Hollande's frequent "escapades".
'President who has a crush'
Gayet is an established television and film actress who also appeared in one of Hollande's 2012 election campaign commercials, in which she described the then-candidate as "marvellous" but also "humble and a really good listener.
She filed a complaint in March over rumours of the affair which she said were a breach of privacy. Her lawyer said at the time there was no basis to
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If confirmed, Hollande's relationship with Gayet would perpetuate a long French tradition of philandering presidents and senior politicians.
Former president Jacques Chirac is believed to have had many extra-marital conquests, as did his predecessor Francois Mitterrand, who even had a daughter born to a mistress.
Valery Giscard d'Estaing was also described as an incorrigible womaniser. He hinted in a recent book at an affair with Princess Diana.
France's media is subject to strict laws on privacy, and has in the past drawn a veil over rumours about the personal lives of the country's leaders.
However, the French are known for being tolerant of their leaders' infidelities, which have in the past proved to have little or no negative impact on popularity ratings.
Laurence Pieau, editor of Closer, told Europe 1 radio that Hollande was just a "normal president" -- a term he himself has used to describe his presidency.
"He's a president who has a crush," she added.