Associated Press/Susan Walsh
President Donald Trump walks down the steps of Air Force One as he arrives a Palm Beach International Airport in West Palm Beach, Fla., Friday, Feb. 17, 2017.
Former U.S. ambassador to the United Nations John Bolton, Lieutenant General H.R. McMaster and Lieutenant General Robert Caslen are also being interviewed for the top national security position, White House spokesman Sean Spicer told reporters at Trump's Mar-a-Lago resort in Florida.
Trump, who has been searching for a new national security adviser for about a week, could add a couple more candidates to the list, Spicer said.
Retired general and former CIA chief David Petraeus is no longer a candidate, Spicer said.
The retired four-star general, who resigned as head of the CIA in 2012 after it was revealed he was having an affair with his biographer, had been on a short list for the job after Flynn was let go.
Spicer said Trump's finalists include Kellogg, Bolton, Caslen, who is the superintendent of the U.S. Military Academy at West Point, and McMaster, who holds a senior post with the U.S. Army Training and Doctrine Command.
Former U.S. National Security Agency head Keith Alexander, and former Army chief of staff Ray Odierno were also thought to be in contention for the job.
Flynn, who was Trump's first national security adviser, stepped down after revelations that he had discussed U.S. sanctions on Russia with the Russian ambassador to the United States before Trump took office and misled Vice President Mike Pence about the conversations.
Finding a replacement has been a challenge for Trump.
Retired Vice Admiral Robert Harward, the president's first choice to take over for Flynn, turned down the offer, citing family and financial reasons.
A source familiar with the matter said Petraeus, like Harward, wanted control over staffing decisions within the NSC, and Trump was reluctant to grant that authority.
Trump is spending the weekend at his properties in Florida.
"Will be having many meetings this weekend at The Southern White House," he wrote in a tweet on Saturday morning.
The president spent the morning at the Trump International Golf Club before returning to his Mar-a-Lago resort. A White House official said he would be having meetings and might play golf.
The rocky start for the NSC under Trump has had an impact in other areas.
The White House dismissed Craig Deare, the NSC's senior director for Western Hemisphere Affairs, after receiving reports that he had criticized the president and top aides, Politico reported. An NSC spokesman declined to comment.
(Additional reporting by Steve Holland and John Walcott in Washington; Editing by Paul Simao)