Opinion polls suggest the runaway favourite among the seven primary candidates is Alain Juppé, the 71-year-old mayor of Bordeaux and a former prime minister and foreign minister.
Mr Juppé is the choice of the establishment and the media. He preaches tax-cutting reforms and reconciliation between France and its Muslim community.
Mr Sarkozy's supporters insist French pollsters are under-counting his strength, just as British and American pollsters underestimated the popularity of Brexit and Mr Trump.
The primary is being watched very carefully by members of the French Jewish community.
Mr Juppé is mistrusted by some, not all, French Jews. They fear he may revive the hostile approach towards Israel associated with his mentor, former president Jacques Chirac.
Allegations have been made by some political blogs that Mr Juppé holds closet antisemitic views. These charges have been dismissed by the front-runner's supporters as crude propaganda circulated by the Sarkozy camp.
Friends point to a statement made by Mr Juppé last year to the umbrella French-Jewish organisation Crif. "When I hear the word Jew, it resounds not in my head but in my heart," he said.
Nicolas Sarkozy, now 61, has run a vigorous campaign, which has included forays into Euroscepticism and climate-scepticism, media-bashing and Muslim-baiting.
He is the favoured candidate of a majority of members of the main centre-right party, Les Républicains, but the primary electorate will be wider. Anyone claiming to share centre-right values and willing to pay a small fee can vote.
Many of this larger cast of voters distrust Mr Sarkozy. They have also been influenced by multiple criminal investigations against the former head of state, including an allegation - angrily denied by Mr Sarkozy - that he took money from the late Libyan strongman Muammar Gaddafi before his successful 2007 presidential campaign.
According to the most recent polls, 39 to 43 per cent of probable primary voters plan to choose Mr Juppé, and 26 to 31 per cent favour Mr Sarkozy.
A couple of recent polls even suggest that another former prime minister, François Fillon, is rising fast and could squeeze Mr Sarkozy out of the two-candidate run-off on Sunday week, November 27.
Arguably, this primary is the "real" French presidential election. President François Hollande, and the left in general, remain cripplingly unpopular.
Marine Le Pen, who has around 25 per cent support, seems certain to keep the left out of the two-candidate second round of the official election next April and May.
Faced with a choice next May between Ms Le Pen and a centre-right candidate - even if it is Nicolas Sarkozy - most French voters of the centre-left will feel obliged to vote anti-Le Pen.
