And Senator Jack Reed, a Rhode Island Democrat and West Point graduate, also made clear within hours of Hagel's resignation that he wasn't interested.
Defense analyst Anthony Cordesman said that as Obama approaches the end of his presidency, the Cabinet post is "not particularly desirable" for anyone with broader political ambitions.
"It's very unlikely you will get political visibility or credit for being the secretary," said Cordesman, who works at the Center for Strategic and International Studies. "There are just too many problems and uncertainties."
Among them: questions about the effectiveness of Obama's military campaign against the Islamic State in Iraq and Syria, Russia's continued provocations in Ukraine, tensions between the White House and U.S. Defense Department over closing the Guantanamo Bay, Cuba, detention center, and concerns at the Pentagon over the impact of deep spending cuts.
Hanging over all of those policy concerns is the uneasy relationship between the White House and the Pentagon throughout Obama's six years in office. His first two defense secretaries, Robert Gates and Leon Panetta, have been bitingly critical about White House efforts to micromanage the Pentagon. And Hagel is said to have grown frustrated by the White House's drawn-out policymaking process and lack of clarity in the president's eventual decisions.
Lawrence Korb, a former U.S. assistant defense secretary, said that with just two years left in Obama's presidency, the next Pentagon chief will be hamstrung in efforts to shape the administration's policies and the department's relationship with the White House.
"The clock is ticking in terms of being able to make significant changes," said Korb, a senior fellow at the Center for American Progress, a think tank closely aligned with the White House.
Mindful that the window on Obama's administration is closing, some White House officials are pressing the president to name Hagel's replacement quickly. He could try to have a nominee in time for a previously scheduled meeting Wednesday in Washington with combat commanders and other military leaders, though officials said that as of Monday, Obama has not made a final decision.
Carter, 60, is seen as the leading contender by many people with knowledge of how such decisions are made. He held two top jobs at the Pentagon during Obama's first term, serving first as the undersecretary for acquisitions and technology before being promoted to deputy defense secretary.
"He's a technocrat and a good manager," said Jane Harman, a former California congresswoman and now president of the Wilson Center. "He knows the acquisitions side of the building and has impressed the military with his commitment to the troops."
Work, a retired Marine, was confirmed earlier this year as deputy defense secretary. He also has ties to Obama, having served on the president's Defense Department transition team after the 2008 election.
Administration officials had been pushing Johnson, the Homeland Security secretary, as a candidate for the Pentagon job. Johnson is a favorite in the West Wing, where officials have praised the process he ran to provide Obama with recommendations for executive actions on immigration.
But Johnson's involvement in that effort risked turning his confirmation hearing into a referendum on the president's immigration moves, which Republicans staunchly oppose. A move to the Pentagon would have also pulled Johnson out of DHS just as the agency is implementing the executive actions he spent months crafting.
Despite the smaller-than-expected pool of candidates, analysts say it would still be difficult for many potential nominees to turn down an offer to join the Cabinet.
"Most people, if asked by the president, will respond," Cordesman said. "But when people like Michele Flournoy take themselves out of the running, you do have to say you may not have as many options as normal."
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Associated Press writer Donna Cassata contributed to this report.
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Follow Julie Pace at http://twitter.com/jpaceDC
