Centrist Francois Bayrou didn't make the runoff, but his strong third-place showing in Sunday's first round of balloting could make him a kingmaker if he throws his support behind the law-and-order former interior minister or to the leftist with a chance of being France's first woman president.
Royal, who previously dismissed calls for an alliance with Bayrou, reached out to him Monday, saying she was available for a public dialogue.
"It is my responsibility to make this overture," she said.
Sarkozy, who courted the far right during his campaign and is criticized by many for abrasive language, sought to soften his image as he appealed to voters in the middle, casting himself as "the candidate of openness."
"Openness of spirit is being able to take into consideration the positions of others, the ability to think that others might be right," he said at a rally in an eastern city, Dijon.
Sarkozy, a Hungarian immigrant's son whose tough stance on immigration enrages some voters, said he aimed to create a "fraternal republic where everyone would find his place."
Bayrou, whose centrist Union for French Democracy has traditionally voted with the right in parliament but who as a candidate attracted leftist sympathizers, kept quiet about endorsing either runoff candidate.
He was not expected to speak out before Wednesday. Sarkozy headed into the runoff with a strong edge in opinion polls and a large pool of votes on the far right that should back him in the May 6 runoff. But a combative Royal insisted the race isn't over. "It's doable," she said. Whoever wins will take the helm of a nation troubled by the 2005 riots of poor young blacks and Arabs, decades of high unemployment, increased economic competition from the likes of India and China and a sense France is waning in global influence. Both candidates are dynamic campaigners - and they are divisive as many French people seem to long for change but can't agree on what needs to be done. Sarkozy calls for relaxing cherished labor protections and cutting taxes to kick start the economy and takes a tough line against illegal immigrants. Royal wants to boost government spending and preserve job protections, while warning against the "brutal" nature of unfettered capitalism. The fight has revived the French appetite for democracy that 12 unimpressive years under conservative President Jacques Chirac had dulled. The 84 percent turnout by voters Sunday topped the usual European range of 50 percent to 80 percent - and was well above the 60 percent U.S. turnout when President Bush was re-elected in 2004. While the feeling that Chirac's departure offers an opportunity for change drew many to the ballot box, so did a desire to avoid a repeat of the shame many people felt when far-right leader Jean-Marie Le Pen reached the runoff in 2002 on a record-low turnout of 72 percent. Le Pen lost five years ago, but the experience reminded voters that their ballots count. This year, voters handed Le Pen a weak fourth-place among a dozen candidates, his second-worst finish in five tries at the presidency. Final results Monday gave Sarkozy 31 percent of the vote and Royal just under 26 percent. Bayrou had 18.6 percent and Le Pen a bit over 10 percent. Two polls after Sunday's vote put Sarkozy in the lead. A CSA poll of 1,005 people had him at 53.5 percent and Royal at 46.5 percent. An IFOP poll gave him 54 percent and her 46 percent. Neither provided a margin of error, but it is generally plus or minus three percentage points in polls of that size. The opinion soundings mean winning over Bayrou's voters will be crucial for Royal. His appeal to follow a middle course and shake up a political system that has been divided between left and right since the French Revolution struck a chord with many voters, who blame the traditional parties for France's woes. "Bayrou has the keys in his hands," said Jean Chiche, an analyst with IFOP, a polling firm. Political analyst Dominique Moisi agreed: "There are very few 'extra troops' so to speak for Segolene Royal. The only thing she can dream for is to turn this election into a referendum against the personality of Nicolas Sarkozy." Both Sarkozy and Royal, a military officer's daughter, are in their 50s and offer France a new generation of politics with no memory of World War II. They drew tens of thousands of young new activists to their parties. They use the Internet as an integral part of their campaigns and they have courted rock stars and celebrities. To secure the Socialist nomination, Royal skirted the party apparatus and its heavyweights - all men - by reaching out directly to supporters. She opened a Web site and encouraged people to come forth with ideas and complaints to be incorporated and addressed in her manifesto. The two finalists are to spar in a televised debate - a tradition between rounds that was skipped in 2002 by Chirac, who refused to appear with Le Pen. The proposed May 2 date for the debate remained unconfirmed. (AP) Associated Press writer John Leicester contributed to this report. AP-CS-04-23-07 1539EDT