President Sarkozy has not yet confirmed that he will stand in next year's April election, but there is now talk of other potential candidates who could run instead of him. These include the current foreign minister, Alain Juppé, and the prime minister, François Fillon.
The poll asked supporters of Sarkozy's party, the UMP, who should be the party's candidate. While 54 percent still named Sarkozy, 18 percent said Juppé, 15 percent Fillon and 6 percent the party leader Jean-François Copé.
When supporters of all parties were asked, 26 percent thought Juppé would be the best candidate for the right, with only 21 percent favouring the current president.
The head of polling organization Viavoice, François Miquet-Marty, said these figures were not reassuring for the head of state.
"The incumbent should have the support of three-quarters of his party," he said. "Having barely the majority is very little."
President Sarkozy has been battling with a wave of bad news and scandals over recent weeks.
The financial crisis and emergency budget measures taken to preserve France's AAA rating in August were compounded by the fall of the French parliament's upper house, the Senate, to the Socialists in elections in late September.
A string of scandals that have touched senior members of the president's entourage are also taking their toll (see below). Some members of the UMP are now talking about replacing the president, although they prefer to remain anonymous for the time being.
"Juppé is gaining ground, it's unquestionable," said one, according to Libération. "He would be ahead of Fillon if party members chose to act. The dynamic is on his side."
In public, party members are resolutely loyal to the president. Juppé himself said in a TV interview last week that he supported the president although he added, mischievously, that he still "dreams" but wouldn't say "what about."
The scandals that have rocked the presidency
The former accountant of L'Oreal heiress Liliane Bettencourt claimed that France's richest woman provided illegal donations to help Sarkozy's election campaign. The charges have been denied by the campaign treasurer and former minister, Eric Woerth.
Kickback payments from arms sales to Pakistan were alleged to have helped fund the campaign of Edouard Balladur, who challenged Jacques Chirac in 1995 for the party's presidential nomination. Sarkozy was Balladur's campaign manager and spokesman at the time. When Chirac won the presidency and cancelled secret payments, a bomb attack in Karachi, Pakistan which killed 11 French engineers, was alleged to be revenge.
An African lawyer and adviser to the president, Robert Bourgi, has claimed that suitcases filled with cash were delivered to various French politicians, including Jacques Chirac and former prime minister Dominique de Villepin, which the latter has strongly denied claiming the charges are politically motivated. Journalist Pierre Péan claims in a book that the payments continued to Mr Sarkozy, which the president's spokesman has also rubbished.
Former finance minister and new IMF president Christine Lagarde is being formally investigated by the ministerial watchdog, the Court of Justice of the Republic, over her role in a €400 million ($560 million) settlement with controversial tycoon Bernard Tapie over allegations of a mishandled sale of his sportswear brand Adidas.