New fathers in France are allowed eleven days of paternity leave by law, increasing to 18 days for multiple births.
During the recent elections to choose a Socialist nominee for the 2012 presidential elections, one of the candidates, Ségolène Royal, proposed giving Sarkozy "paternity leave of five years."
It is unlikely the president will take any days off. He has a packed agenda over the coming days and weeks. On Wednesday, he left his wife in the La Muette clinic and flew to Frankfurt to meet German chancellor Angela Merkel to discuss a deal to save the euro.
He will also attend a summit on Sunday in Brussels which is planned to adopt a comprehensive deal to solve the ongoing sovereign debt crisis.
His reluctance to take the time off may not be only due to a busy schedule, but also a macho political culture in France.
According to sociologist Christine Castelain-Meunier, quoted in Thursday's Parisien newspaper, the reluctance to take paternity leave is common in the political classes.
"Since it was created in 2002, no French male politician of any note has publicly taken his paternity leave," she said. "Perhaps some have taken it but not wanted to be blacklisted for it."
Senior politicians in other countries have been more willing to take their allotted leave. Prime Minister David Cameron took two weeks of paternity leave when his wife, Samantha, gave birth in 2010, as did his rival, opposition leader Ed Milliband.
In Norway, the minister of children, equality and social cohesion took a full sixteen weeks of paternity leave.
Franck Louvrier, one of the president's advisers, told the newspaper that time off was highly unlikely.
"There is one person who is exempt from parental leave and that's the head of state," he said.
Other new fathers in France are more willing to take advantage of the time off with two-thirds taking full paternity leave.
Employers' union, Medef, has called for parternity leave to become obligatory.