But the publication marked the latest incursion into the once-sacred private lives of French politicians. The tradition of keeping private lives private has been chipped away since Hollande's predecessor Nicolas Sarkozy divorced his wife Cecilia, who was having an affair, and remarried model and singer Carla Bruni.
Bloggers, tweeters and other online sources have tapped into public curiosity and pushed the borders of French privacy. Closer last made a splash when it published topless photos of British Prince William's wife, Kate.
Hollande said in a statement he "deeply deplores the attacks on respect for privacy, to which each citizen has a right." He is weighing possible action, including via the courts, it said. He did not address the alleged affair.
Hollande, who was dubbed "Monsieur Normal" by French media, has never married. He had four children with politician Segolene Royal, and has been living in recent years with his partner, journalist Valerie Trierweiler, considered France's first lady.
Gayet's lawyer and agent did not immediately return phone and email messages Friday about the Closer report. The actress, who has performed in nearly four dozen films and sometimes appears on talk shows, praises Hollande in the campaign spot for his humility and character, and says he is someone who "really listens."
Closer published images it says show one of Hollande's bodyguards at the door to Gayet's building, and a motorcyclist arriving to drop off a helmeted man on the night of Dec. 30 and escorting him away the next morning. The magazine, a tabloid known for suggestive photos and gossip, says the alleged affair raises questions about the president's security, if he is sneaking out with a single bodyguard and traveling exposed on the back of a motorcycle.
It is unclear how a potential romantic scandal might affect Hollande's presidency. His popularity is already very low, largely over voter frustration at his failure to reduce unemployment and get the economy growing.
Last year, according to French media reports, Gayet filed a lawsuit accusing some websites of invasion of privacy over reports on the alleged affair. Her lawyer issued a statement to AFP at the time saying that she was targeting "a rumor spread on the Internet that attributes to her, without any basis, a romantic relationship with the president."
The magazine says it conducted an investigation and claims that "there is, between Francois Hollande and Julie Gayet ... a true passion that has turned their hearts upside down, rocked their lives and made them take foolish risks."
It says the apartment is in the same Paris district as the president's palace, and is Gayet's but registered in the name of actor friends. It says the pair has been meeting there secretly since June.
Associated Press writers Thomas Adamson and Sylvie Corbet contributed to this report.