The two rivals in France's presidential race have been trying to woo supporters of centrist candidate Francois Bayrou, who came third on Sunday.
Ms Royal and Mr Sarkozy are locked in battle for the middle ground
The Socialist candidate, Segolene Royal, made a direct appeal to him to form an alliance. "I'm waiting for a response," she said.
Nicolas Sarkozy's team accused her of putting power before principles.
Mr Sarkozy of the centre-right UMP is favourite to win the run-off. His aides want a deal with Mr Bayrou's MPs.
Ms Royal will face Mr Sarkozy in the run-off on 6 May and opinion polls point to a victory for Mr Sarkozy.
Ms Royal said she was "not exerting any pressure" on Mr Bayrou "or setting him any ultimatum".
But she added: "I think it is my responsibility as a presidential election candidate to work out which ideas we can agree on".
The votes of Mr Bayrou's supporters are likely to prove decisive
Nearly seven million voters backed Mr Bayrou, leader of the UDF, in the first round.
He will give an eagerly awaited news conference at 1530 (1330 GMT) on Wednesday, but allies say he is unlikely to back either candidate openly. He is seen as a potential "kingmaker" who could make or break the election for either side.
The UDF has joined conservative governments in the past and some of Mr Sarkozy's aides are exploring a possible deal again for the forthcoming parliamentary elections in June.
But in the run-up to the first round Mr Bayrou stressed his desire for a third way in French politics, to jettison the traditional left-right split.
For his part, Mr Sarkozy told supporters in Dijon, eastern France: "I will not cut a deal at the expense of my convictions."
Mr Sarkozy won 31% of Sunday's vote, while Ms Royal, bidding to be France's first female leader, took nearly 26% - beating 10 other candidates.
Opinion polls gave Mr Sarkozy between 52 and 54% of the vote in the second round - against 46 and 48% for Ms Royal. Only 14% were uncertain of who they wanted to cast their ballot for.
They will confront each other in a televised debate on 2 May.