"The president told us in cabinet that he would immediately submit a new draft if there is a rejection by the Constitutional Council" of a bill approved recently by the French parliament, said one of the ministers, who spoke on condition of anonymity.
Another minister said Sarkozy had also criticised those in cabinet who had opposed the bill, saying they "did not see past the ends of their noses".
He said a rejection of the bill by the Constitutional Council could open the door to questioning a law that penalises denial of the Holocaust.
After being approved by the National Assembly and Senate, the law was put on hold on Tuesday after politicians opposed to the legislation demanded that its constitutionality be examined.
Two separate groups of French politicians who oppose the legislation -- from both the Senate and the lower house -- said they had formally requested the Constitutional Council examine the law.
The groups said they each had gathered more than the minimum 60 signatures required to ask the council to test the law's constitutionality.
The council is obliged to deliver its judgement within a month, but this can be reduced to eight days if the government deems the matter urgent.
Turkey reacted furiously last week when the Senate approved the law, which threatens with jail anyone in France who denies that the 1915 massacre of Armenians by Ottoman Turk forces amounted to genocide.
Despite government backing of the law, at least two ministers, Foreign Minister Alain Juppé and Agriculture Minister Bruno Le Maire, had spoken out against the bill.
France has already officially recognised the killings as a genocide, but the new law would go further by punishing anyone who denies this with up to a year in jail and a fine of €45,000 ($57,000).
Armenians say up to 1.5 million of their forebears were killed in 1915 and 1916 by the forces of Turkey's former Ottoman Empire.
Turkey disputes the figure, arguing that 500,000 died, and denies this was genocide, ascribing the toll to fighting and starvation during World War I and accusing the Armenians of siding with Russian invaders.
Ankara has already halted political and military cooperation with France and was threatening to cut off economic and cultural ties, while hailing the move to take the matter to the Constitutional Council.
Trade between the two states was worth €12 billion ($15.5 billion) in 2010, with several hundred French businesses operating in Turkey.