Mr Trump, a property tycoon and Republican outsider, won an unexpected victory against Hillary Clinton.
He has already replaced New Jersey Governor Chris Christie with Vice-President-elect Mike Pence as head of the transition team.
Media reports say Mr Trump’s son-in-law and close adviser Jared Kushner was behind the change.
Mr Christie was New Jersey attorney general when Mr Kushner’s father was tried and jailed in the state for tax evasion, illegal campaign contributions and witness tampering in 2004.
Former Congressman and House intelligence committee chairman Mike Rogers, who was handling national security for the transition, announced on Tuesday that he was leaving.
He and another member of the national security team, Matthew Freedman, were sacked, according to the New York Times.
Mr Rogers is thought to have been close to Mr Christie, while Mr Freedman is said to be a protege of Paul Manafort, Mr Trump’s former campaign manager who quit in August.
But Mr Trump sought to calm fears of turmoil.
“Very organized process taking place as I decide on Cabinet and many other positions,” he tweeted.
“I am the only one who knows who the finalists are!”
A break with protocol: By David Willis, BBC News, Washington
Donald Trump has spent the week since his election holed up in the Manhattan skyscraper that bears his name.
He’s been looking to fill key posts in his cabinet, but the early indications are it’s proving a less- than-straightforward task.
There have already been calls for Mr Trump to rescind one of the appointments he has made – that of the former head of the right wing Breitbart website Steve Bannon as senior White House adviser.
Following a day of meetings, and a reassurance from his press spokesman that he would be staying in for the evening, Mr Trump took to a New York steak house for dinner with his family last night – a break with protocol which left some reporters speculating that he might not be fully comfortable with the sort of scrutiny that comes with the presidency.
The former mayor of New York, Rudolph Giuliani – who has been tipped for a senior post – said presidential transitions were always a complex process, and glitches were normal.
Mr Trump is due to be inaugurated as president on 20 January 2017.