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Washington (CNN) -- With the no-fly zone now in place in the skies over parts of Libya, the hastily-assembled coalition of nations enforcing it is straining under decisions about next steps, and who should be in the lead.
On Tuesday, President Barack Obama called French President Nicolas Sarkozy to review the situation in Libya, with the White House reporting they "agreed on the means of using NATO's command structures to support the coalition" but giving no immediate details.
Obama also spoke with British Prime Minister David Cameron, according to a Downing Street spokesman who said the two leaders agreed "that NATO should play a key role in the command structure going forward, and that these arrangements now needed to be finalized."
And speaking Tuesday on a visit to El Salvador, Obama said he had "no doubt" that the United States will be able to shift control of the Libyan military mission to an international coalition, and that the timetable for such a transition continued to be in coming days, rather than weeks
Obama said that once leadership of the military mission in Libya shifts from the United States to the coalition, "it is not going to be our planes maintaining the no-fly zone" and "it is not going to be our ships that are necessarily involved in enforcing the arms embargo."
The international support for the military mission, with NATO allies and Arab nations taking part, meant that "the United States is not going to be bearing all the cost," he said.
But there has been intense debate among the allies over NATO's role going forward.
Up to now, the United States has closely coordinated military action with its Western allies. But the Obama administration, already embroiled in waging and paying for two wars, wants to exit that role as soon as possible.
The natural choice would seem to be NATO with its experience leading and coordinating international coalitions, but U.S. and European officials tell CNN that some allies, especially France's Sarkozy, are concerned the action in Libya could be perceived negatively as another NATO operation in another Arab country.
It would be too easy, they say, for Arabs who don't like the west to label this as "crusade." Sarkozy wants to get the political language on NATO's involvement right, and U.S. and U.K. officials are sympathetic with this.
French officials Tuesday publicly presented the outlines on a solution for who should lead: a tandem approach that combines political leadership with NATO's assets and capabilities. That dual approach seems aimed at avoiding the diplomatic pitfalls of leadership role for NATO.
Foreign Minister Alain Juppe proposes creating what he calls a "political steering committee" that would include the foreign ministers of Western as well as Arab members of the coalition. He said the British are in agreement on the proposal which he said he made at the request of President Sarkozy.
Juppe said he wants the committee to hold its first meeting in the next few days in either London, Brussels or Paris and to hold regular meetings after that in the same cities.
Laurent Teisseire, spokesman for the French Ministry of Defense, gave further details of the proposal at his Tuesday briefing. He said there had been "vivid debate" over the past days but denied it was an "impasse" or a "dead-end" and said diplomats "were converging on a solution which will be effective."
"Basically what we try to differentiate is the political leadership, on one hand, which we consider should be based on nations enforcing the (U.N.) Resolution, plus the Arab League," he said, "and the use of NATO assets and capabilities which are collective too and which we think we should use."
Beyond debate over NATO's role, several Western diplomats and senior U.S. officials tell CNN there also seems to be general annoyance with Sarkozy's need to be publicly out in front.
French fighter jets began the operation to enforce the no-fly zone and officials say there was a general understanding among the allies that the French would be in the air before anyone else. However, nobody was really well informed, they say, on the sequence of events that the planes would take off even as a meeting of foreign ministers, called by Sarkozy, was taking place in Paris Saturday, allowing the French president to unilaterally announce commencement of military action.
In another sign of dissension in the ranks, the French and German ambassadors Monday walked out of a meeting of the North Atlantic Council, the alliance's decision-making body, after Secretary General Anders Fogh Rasmussen criticized the French for hampering NATO involvement and Germany for not actively participating.
Another complication is how to consolidate Arab support. The United States expects additional Arab support for the coalition enforcing the no-fly zone over Libya, a senior administration official told reporters Tuesday.
So far, Qatar is the only Arab country that has contributed planes to mission.
The official said that several Arab states are in the process of finalizing their plans, adding the Obama administration was "confident we will have further concrete contributions of different kinds" for enforcement of the resolution in the next two to three days.
NATO is in the process of finalizing the exact structure for command of the mission, the official said, adding that NATO would play a "key role" as part of a wider effort that would include Arab states and other nations. NATO has a long history of working in partnership with non-members, the official noted, citing missions in Afghanistan and Kosovo.
The official added that the United States will revert to a supportive role in the next few days, which could include jamming of Libyan leader Moammar Gadhafi's communication, refueling and intelligence support.
CNN's Tom Cohen contributed to this report.