WASHINGTON — Wanted: Thick-skinned candidate to oversee a sprawling bureaucracy bitten by budget cuts and join a national security team besieged by criticism. Must be tolerant of White House interference.

The job conditions for President Obama’s next defense secretary have already spurred some top contenders to bow out, leaving the White House with a slim list of candidates to fill the post for the administration’s final two years. On Monday, Homeland Security Secretary Jeh Johnson became the latest to tell the White House he wasn’t interested in the job, according to people familiar with the process.

Johnson’s decision to stay put at DHS is likely to deepen the impression that the Pentagon post — typically a highly sought-after Cabinet spot —is drawing little interest. The president’s short list of contenders now includes longtime public servants who have deep Pentagon experience, but may be less likely to give Obama’s national security agenda the jolt that critics — and increasingly some supporters — say is needed.

Top contenders include Ashton Carter, the former deputy defense secretary who left the administration in late 2013, and Robert Work, who now holds the Pentagon’s No. 2 job.

That’s a far shorter list than some in the White House had hoped when Defense Secretary Chuck Hagel resigned last week under pressure from Obama. Michele Flournoy, one of Obama’s top choices, quickly took her name out of contention, in part because of concerns over the tight rein the White House has tried to keep on the Defense Department. Sen. 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 Defense Department over closing the Guantanamo Bay detention center, and concerns at the Pentagon with 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 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 both 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 seek 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.”