They said Merkel, Europe's most powerful political leader, made the inquiry in a private conversation with French President Francois Hollande after European Parliament elections characterised by widespread anti-EU protest votes.
"Merkel raised it privately with Hollande, who did not take a final position but said he did not think it would be a good idea for Europe to lose the IMF post," one source said.
In Berlin, German government spokesman Steffen Seibert said the Reuters report was wrong. Merkel had reaffirmed on Monday her support for former Luxembourg Prime Minister Jean-Claude Juncker, who led the centre-right European People's Party slate in the parliamentary elections, he said. The centre-right topped the poll but fell well short of an overall majority.
Emerging nations have said they want to break a long-standing arrangement under which a European gets the top IMF job and an American heads the World Bank.
Another French source said there was no way the Socialist president, under pressure from the far-right National Front, which won the election in France, and the left wing of his own party, could back a member of the centre-right opposition for the top European Union job.
Lagarde, 58, was finance minister under former President Nicolas Sarkozy before moving to the IMF at the height of the euro zone's debt crisis in 2011 to replace disgraced French politician Dominique Strauss-Kahn. Her term runs until 2016.
A fluent English-speaker who headed a major U.S. law firm before holding ministerial office in France, she is highly regarded by Merkel and German Finance Minister Wolfgang Schaeuble as a steady-handed compromise builder.
The spokesman referred to Merkel's public statements noting that a qualified majority of member states was necessary and saying she wanted to negotiate in a "European spirit" of seeking the broadest possible consensus.
The chancellor has acknowledged that Juncker may not have sufficient backing among EU leaders to secure their nomination, and that other candidates could also do the job provided the result of the elections was respected. To critics, Juncker represents a political class that has alienated many of Europe's voters and played into the hands of Eurosceptic parties.
In Washington, IMF spokeswoman Conny Lotze said: "As Madame Lagarde has said repeatedly, she is focused on her work at the IMF and intends to complete her term as Managing Director."
German and IMF officials said Merkel had a private meeting with Lagarde during a visit to Washington in early May. They saw each other again in Berlin two weeks later when Lagarde attended a meeting of the heads of major international economic organisations hosted by the German government.
British Prime Minister David Cameron has led opposition to Juncker's bid to succeed former Portuguese Prime Minister Jose Manuel Barroso, arguing that the EU needed new leadership committed to reform in response to voters' dissatisfaction. London sees Juncker as an old-style European federalist.
British officials have made clear that Lagarde would be an acceptable alternative, as would centre-left Danish Prime Minister Helle Thorning-Schmidt. Some British newspapers have campaigned for Lagarde to be given the role.
The leaders of Sweden, the Netherlands and Hungary objected to the principle of making the leading candidate of the biggest party in parliament the nominee, and Italian Prime Minister Matteo Renzi has said Juncker has no automatic right to the job.
EU leaders mandated European Council President Herman Van Rompuy to propose by late June a package deal of senior EU appointments and a draft policy agenda for the new EU executive, which is due to take over in November.
Diplomats said Merkel, Hollande, Cameron, Renzi and Van Rompuy may discuss the issue on the sidelines of a Group of Seven nations summit in Brussels on Wednesday and Thursday.
The Commission candidate nominated by EU leaders must win the backing of a majority of the 751-member European Parliament, and some lawmakers have threatened to vote against anyone who was not a candidate in the election.
That raises the possibility that if the leaders do not put Juncker forward, their nominee may be voted down.
The sources said Hollande had not yet decided who to put forward for the EU carousel, but if Merkel was open to a French candidate for one of the top jobs, Paris might offer Socialist former Prime Minister Jean-Marc Ayrault, a fluent German-speaker who was replaced in March but is well regarded in Berlin.
Other posts in the mix include the president of the European Council, who chairs EU summits, the bloc's foreign policy chief, a possible permanent chairman of the Eurogroup of euro zone finance ministers, and the president of the European Parliament.
The package needs to balance left and right, north, south, east and west, large and small states, and men and women.