NORTH CHARLESTON, S.C. (Reuters) - U.S. President Donald Trump, scrambling to find a new top security aide after firing his first one and being spurned by another candidate, said on Friday he has four people under consideration including acting national security adviser Keith Kellogg. Trump ousted Michael Flynn on Monday in a controversy over the retired lieutenant general’s contacts with Russia. Retired Vice Admiral Robert Harward on Thursday turned down the Republican president’s offer to replace Flynn. “General Keith Kellogg, who I have known for a long time, is very much in play for NSA - as are three others,” Trump said on Twitter, without naming the other candidates. Former CIA chief David Petraeus was previously identified as a candidate by a White House official. Former U.S. National Security Agency head Keith Alexander and former supreme allied commander in Europe James Jones, who held the national security adviser post under former Democratic President Barack Obama from 2009 to 2010, were also thought to be under consideration. Both are retired generals. Two others also thought to be in contention were former U.S. ambassador to the United Nations John Bolton and Lieutenant General H.R. McMaster, who holds a senior post with the U.S. Army Training and Doctrine Command. Kellogg, a retired lieutenant general who is currently chief of staff of the White House National Security Council, accompanied Trump on a trip to South Carolina on Friday before heading to Florida. He stepped into the national security adviser role on an acting basis after Flynn’s firing. Trump may meet with candidates for the post during his weekend visit to Florida, a White House official told reporters. Petraeus held command posts in the wars in Iraq and Afghanistan and served as CIA director under Obama. He quit as CIA chief in 2012 and pleaded guilty to a misdemeanor charge of mishandling classified materials relating to documents he had given his biographer, with whom he had an affair. Harward, a senior executive at Lockheed Martin and former Navy SEAL, declined Trump’s offer in part because he wanted to bring in his own team, according to two sources familiar with Harward’s decision. The White House chief of staff, Reince Priebus, told Fox News on Friday that Harward’s family “didn’t sign off” on him taking the job. “That’s all it is,” Priebus said. Richard Haass, who held senior White House and State Department posts under Republican presidents and now heads the Council on Foreign Relations, said on Twitter the new national security advisor should insist on the right to choose staff members and have unlimited access to the president. Haass, who Trump considered for a job in his administration, also called for rescinding a directive from the president that gave Trump’s chief White House strategist, Steve Bannon, a seat on the National Security Council, a move condemned by Democrats. Trump’s administration has been dealing with the fallout from Flynn’s departure for much of the week. Flynn, a close adviser to Trump during his presidential campaign last year, was seen by Moscow as a leading advocate of friendlier ties with Russia. Trump said on Thursday he fired Flynn because he had misled Vice President Mike Pence about conversations he had with the Russian ambassador to the United States, before Trump took office, about sanctions imposed by Obama’s administration. Trump has defended Flynn’s actual contact with the ambassador, saying what he did “wasn’t wrong.” The Washington Post reported on Thursday that Flynn told FBI agents last month that he had not discussed sanctions with the ambassador. Flynn’s Jan. 24 interview with the FBI could expose him to charges, since lying to the agency is a felony, but any decision to prosecute would lie with the Justice Department. 