Mrs Bruni-Sarkozy, whose past lovers include Mick Jagger and Eric Clapton, resented Miss Dati's friendship with the president, the book claims.
Following a New Year's Eve dinner at the Elysée, Mrs Bruni-Sarkozy and Miss Dati were walking in the palace's private apartments when the then Miss Bruni is said to have pointed at Mr Sarkozy's bed and said: "You'd have loved to occupy it, wouldn't you?"
The book says the women "who were just getting to know each other, were also learning how to detest each other".
Miss Dati was a close friend of Mr Sarkozy's ex-wife, Cécilia Ciganer-Albéniz, who called her "my little sister" - making the new Mrs Bruni-Sarkozy dislike her even more.
Miss Ciganer-Albéniz, who has since married Richard Attias, an events planner, has made some claims of her own about the women in her former husband's life.
Carla BruniIn Cécilia, a book by the French journalist Anna Biton, Miss Ciganer-Albéniz called the president's female friends a "bunch of slappers" and young female government ministers "boring wallflowers".
She also branded Mr Sarkozy a "stingy philanderer" with a "behavioural problem" who is an "unworthy president" of France.
Recalling her first meeting with Mr Sarkozy - who is some almost five inches smaller than his statuesque first lady - Mrs Bruni-Sarkozy says: "I didn't expect him to be that funny, and so lively."
The dinner party was the home of public relations Jacques Séguéla, with Mrs Bruni-Sarkozy saying: "When I arrived, I realised that it was a blind date.
Carla BruniReferring to the ancient Greek myth of Sisyphus, a king punished by being cursed to roll a huge boulder up a hill for eternity only to watch it roll down again, Mrs Bruni-Sarkozy said Mr Sarkozy is: "Like all of us, a bit like Sisyphus, he likes to push the rock. But he's made of stern stuff. Three rays of sunshine and he finds life magnificent."
On Wednesday both the Elysée Palace and Ministry of Justice in Paris said they had no comment to make about the love lives of senior ministers, let alone the president.
But government sources pointed to extracts in the book in which the frosty relationship between Mrs Bruni-Sarkozy and Miss Dati is played down.
It points out that Mrs Bruni-Sarkozy, a multi-millionaire in her own right, helped Miss Dati during her campaign to become mayor of Paris's VIIth arrondissement.
Referring to Miss Dati, Mrs Bruni-Sarkozy says: "I see her frequently, she makes me smile a lot. The rumours must come from claims that she's a friend of the ex-wife of my husband.