From Elise Labott and Paula Newton, CNN
NEW: Clinton says NATO is drawing up plans for possible broader command
The deal means NATO will assume command of one part of the military mission
NATO is still considering whether to also take command of the rest of the mission
The agreement followed talks involving Britain, France, Turkey and the United States
(CNN) -- NATO agreed Thursday to take command of enforcing a no-fly zone over Libya and was considering taking control of the full U.N.-backed military mission, Secretary-General Anders Fogh Rasmussen told CNN.
Rasmussen's announcement fell short of what U.S. President Barack Obama has sought, and it was unclear if concerns by Turkey and some other NATO allies over coalition airstrikes on Libyan ground forces would prevent NATO from agreeing to expand its command over the entire mission.
"What we have decided today is that NATO will enforce the no-fly zone," Rasmussen said in an interview with CNN's Wolf Blitzer. "We are considering whether NATO should take on overall responsibility. That decision has not been made yet."
Rasmussen said he expected NATO to assume command of the no-fly-zone enforcement in a few days, and to decide on the issue of broader responsibility "within the coming days."
Asked if the announcement revealed a split in NATO over the mission, Rasmussen said no. However, he also acknowledged that if unaltered, the agreement would mean the overall Libyan mission would have two parts, with NATO enforcing the no-fly zone and the U.S.-led coalition that launched the mission handling a naval blockade and airstrikes.
The deal was reached after a conference call earlier in the day between U.S. Secretary of State Hillary Clinton and her counterparts from Britain, France and Turkey, according to diplomatic officials who spoke on condition of not being identified by name.
In a statement to reporters later Thursday after Rasmussen's announcement, Clinton said all 28 NATO allies authorized military authorities to develop a plan for NATO to take on the broader mission of civilian protection under the U.N. Security Council resolution authorizing the military mission.
Clinton also said she will travel to London on Tuesday to attend an international meeting on Libya that will include NATO allies and Arab partners in the Libya mission. She took no questions.
At NATO headquarters, Thursday's meeting extended long past its expected conclusion. NATO sources said a major sticking point involved the rules of engagement for coalition forces that will take over command of enforcing a no-fly zone and naval blockade of Libya.
When Rasmussen finally emerged to announce an agreement, it was clear that questions over the rules of engagement remained unresolved.
He said NATO would use the already established chain of command for enforcing the no-fly zone. The NATO supreme commander, an American, would be in charge, but the mission would be under NATO control, Rasmussen noted.
In addition, non-NATO partners including Arab countries would participate, Rasmussen said.
NATO sources told CNN that the shift in command for enforcing a no-fly zone was expected by Sunday night.
According to the sources, NATO has sent a directive to NATO's military chain of command asking for a plan on how to execute an expanded role for enforcement of U.N. Resolution 1973. Under the expanded role, called "no-fly plus" by some officials, NATO might be given more robust rules of engagement to ensure that civilians are protected, the sources said.
One proposal for "no-fly plus" would allow some coalition forces to withdraw from certain missions, the sources said.
So far, U.S. forces have taken on the bulk of the Libyan mission, according to figures provided by the Pentagon. Of a total of 175 Tomahawk missiles fired, 168 were from the United States and seven from Great Britain, the only two countries to possess them, while U.S. planes have flown almost two-thirds of the sorties and U.S. ships comprise more than two-thirds of the total involved.
Obama has repeatedly said that the United States will turn over control of the Libya mission to allies within "days, not weeks." While he specified that would mean no U.S. planes in the air over Libya, Obama left open the possibility of U.S. naval vessels playing a role in blockading Libya against arms shipments.
The U.N. Security Council resolution authorizing the mission includes a section that allows coalition forces to take other steps as necessary to protect Libyan civilians. So far, the U.S.-led coalition has interpreted that to include airstrikes on Libyan ground forces threatening the rebel stronghold of Benghazi and in other areas.
NATO sources said Turkey was uncomfortable with such a role and raised concerns about that at Thursday's meeting.
CNN's Jennifer Rizzo, Larry Shaughnessy, Tom Watkins and Tom Cohen contributed to this story.