"I will not forge an alliance or strike a deal behind the voters' backs. I want a clear vote. I want a transparent political debate," she told RMC radio.
Michel Rocard, the former socialist prime minister, last week raised the idea of having Bayrou and Royal appeal publicly to supporters to back the other in a runoff against Sarkozy.
Twelve candidates will contest the vote on April 22 and if, as looks almost certain, none of them wins an absolute majority, the top two will go forward to a second round ballot two weeks later.
Mar 16: Deadline for presidential candidates to get at least 500 mayors or other elected representatives to sponsor them. Without these, candidates cannot stand.
Mar 19/20: Constitutional Court examines proposals, decides list of official candidates.
Apr 9-20: Official campaign period.
Apr 21: First round presidential voting in overseas territories and departments. Guadeloupe, Martinique, Guyana, Saint-Pierre-et-Miquelon and French Polynesia.
Apr 22: First round of voting on mainland France, the Indian Ocean islands of Reunion and Mayotte and in New Caledonia.
May 5: Second round of overseas voting.
May 6: Second round of mainland voting.
May 17: End of President Chirac's mandate.
Jun 10: First round of parliamentary voting.
Jun 17: Second parliamentary round.
Jun 19: End outgoing parliament.
Surveys have suggested that Bayrou - unlike Royal - would stand a good chance of beating the former interior minister if he made it to the second-round.
"For left-wing voters, the priority is to defeat Sarkozy," Roland Cayrol, director of the CSA polling institute, said. "Their problem is finding the most effective vote to do so."
Bayrou has positioned himself as a candidate who rejects the left-right divide, but since its creation in 1978 his Union for French Democracy party has been in almost permanent alliance with the right.
At least one third of the electorate remains undecided but the most recent surveys have shown Sarkozy maintaining his lead with Royal rallying support.
Bayrou, who has been campaigning to break the grip on French politics by the centre-right UMP party and the Socialists, has seen his support slip.
Concerns over jobs and security have dominated the campaign but the final choice for many voters is likely to come down to personality rather than policy.
Sarkozy, seen as the most economically liberal of the contenders, is admired by supporters for his crackdown on rioting in France's poor suburbs in 2005 and for his tough line on illegal immigration and crime.
But many voters consider him dangerously authoritarian and are incensed by his plans for a ministry of immigration and national identity - a project they say is aimed at taking votes from the far-right National Front.
Royal, the first woman with a serious chance of winning the French presidency, has put forward a left-wing economic platform combined with a strong emphasis on traditional social values.
Initially celebrated as a refreshing change in French politics, her campaign has been stalled by a series of gaffes and questions over ambiguous policy proposals.