French newspapers were prompt in front-paging the couple's pictures. Sources said the president's office would not comment on the status of the two.
Daily Le Figaro, considered close to the conservative president, ran the image of the Italian-French Bruni on Monday, headlining 'Carla Bruni: The President's Girlfriend".
Sarkozy and Bruni visited Disneyland on Saturday, when the park was thronged with visitors. Several photographers were reportedly on hand and openly snapped the pair, who seemed at ease with the attention.
Sarkozy, 52, basked in the media spotlight in his long race to the presidency leading up to his May election. He bristled, however, at intense media attention to the end of his tempestuous marriage to Cecilia Sarkozy, a former model. The two divorced two months ago.
Bruni, 39, a supermodel who won critical and public acclaim for her first folk album in 2003, has been linked in the past to a range of famous men, from rockers Mick Jagger and Eric Clapton to real estate mogul Donald Trump and actor Kevin Costner.