Admitting defeat, Mr Sarkozy endorsed Francois Fillon, a moderate who finished first in Sunday`s first round, according to near-complete results.
Alain Juppe, who like Mr Fillon is an ex-prime minister, finished second.
They will face each other in a run-off next Sunday. The winner will compete in next year`s presidential election.
The winner of the Republican primary is likely to make the presidential run-off, where he or she will probably face far-right leader Marine Le Pen.
With the governing socialists unpopular and divided, it seems unlikely that any left-wing candidate will survive the first round in April.
Polls currently suggest that the centre-right candidate would win the second round in May.
In his concession speech, Mr Sarkozy, 61, said: "I have no bitterness, I have no sadness, and I wish the best for my country."
He told reporters he now supported Mr Fillon, 62, whose "political choices" he said were closer to his own than Mr Juppe`s.
Mr Fillon served as Mr Sarkozy`s prime minister during the latter`s presidency between 2007 and 2012.
France centre-right primary: Ex-President Nicolas Sarkozy knocked out
21 November 2016
Read: 1121
French ex-President Nicolas Sarkozy has been knocked out of a primary to choose the presidential candidate of the centre-right Republican party.
-1780923278.jpg&h=190&w=280&zc=1&q=100)
