With Michael Flynn out and Vice Admiral Robert Harward turning down the job, many are asking: Who will be President Trump’s new national security adviser?

There are two other names rumored to be front-runners: Retired Lt. Gen. Keith Kellogg and retired Gen. David Petraeus.

Mr. Trump said in a post on Twitter on Friday he was weighing four potential candidates for national security advisor.

"General Keith Kellogg, who I have known for a long time, is very much in play for NSA – as are three others," he tweeted, without naming the other candidates.

Lt. Gen. Kellogg was named as acting National Security Advisor following Mr. Flynn's resignation on February 13. Before taking over for Flynn, Kellogg was named chief of staff and executive secretary of the National Security Council, an important supporting role. Kellogg was in the military for 36 years, including serving in the Army in Vietnam and as a Special Forces officer in Cambodia. He also served as chief operation officer for Baghdad's provision government in the early 2000s.

Trump is also reportedly considering General Petraeus, whom he previously considered for secretary of State before giving the position to Rex Tillerson, the former chief executive officer of oil and gas multinational ExxonMobil. During his almost four-decade tenure in the US Army, Petraeus served as Commanding General of coalition forces in Iraq from 2007 to 2008. In 2011, Petraeus was confirmed unanimously by the Senate as CIA director. But he resigned in 2012, following an extra-marital affair with his biographer Paula Broadwell, during which he had given Ms. Broadwell access to classified information.

Two others reportedly being considered for the national security adviser post are Gen. James Jones, the former supreme allied commander in Europe who served as national security adviser to Obama from 2009 to 2010; and Gen. Keith Alexander, former National Security Agency chief.

Harward, a former Navy SEAL and deputy commander of US Central Command under Trump’s current defense secretary, Gen. James Mattis, issued a statement Thursday confirming the nomination, and the explained why he felt he needed to turn the position down.

The Trump administration was “very accommodating to my needs, both professionally and personally,” Harward told the Associated Press. “It’s purely a personal issue for me. I’m in a unique position finally after being in the military for 40 years to enjoy some personal time.”

A friend of Harward’s told CNN that the retired vice admiral turned down the position because of the chaotic nature of Trump’s White House.

On Friday, meanwhile, the White House chief of staff, Reince Priebus, told Fox News that Harward's family "didn't sign off" on him taking the job.

"That's all it is," Priebus said.

And as NPR’s Pentagon correspondent Tom Bowman noted, taking over the role of NSA would be an especially challenging position in the Trump administration.

“You have a president that doesn’t have a lot of background in national security or really governing. And then you have people freelancing, like political strategist Steve Bannon, who had a role in writing the immigration ban,” said Mr. Bowman. “So it’s going to be hard for someone to take that job, very tough.”

Harward would have replaced Flynn, whom Trump appointed as his national security adviser on Nov. 18. After 23 days on the job, Flynn resigned amid claims that he discussed US sanctions with the Russian ambassador to Washington, Sergey Kislyak, the same day the Obama administration announced the sanctions.

Vice President Mike Pence defended Flynn in early January, saying that the phone calls between Flynn and Ambassador Kislyak on Dec. 29 had nothing to do with the US sanctions. Resigning on Monday, Flynn admitted that he had “inadvertently briefed the vice president-elect and others with incomplete information.”

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Harward’s pass on the position further puts a fire under the Trump administration to find a replacement – and fast, as The Christian Science Monitor’s Linda Feldmann noted Wednesday:

Mr. Trump’s immediate challenge is to put his presidency back on track, and that means quickly replacing Mr. Flynn with a seasoned professional who plays well with others, analysts say. Only then can the Trump administration begin to fully address all the foreign policy and national security matters on its plate. The position of national security adviser is a linchpin for foreign policy, working in close proximity to the president. Past occupants of the job, Henry Kissinger (who advised Presidents Nixon and Ford), Brent Scowcroft (who advised Presidents Ford and George H.W. Bush), and Zbigniew Brzezinski (who advised President Carter) are all giants of foreign policy thought.

This report contains material from the Associated Press and Reuters.