Mr Trump tweeted that the process of selecting his new cabinet and other positions was "very organised".
US media say two senior members of the transition team working on national security have been forced out.
Mr Trump, a property tycoon and Republican outsider, won an unexpected victory against Hillary Clinton.
Outgoing President Barack Obama, on a visit to Greece, admitted he and his successor "could not be more different" but vowed to "do everything we can to support the smoothest transition possible," saying democracy depended on it.
Mr Trump has already replaced New Jersey Governor Chris Christie with Vice-President-elect Mike Pence as head of his 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.
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.
And Mr Trump himself 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!"
He said that contrary to a report in the New York Times, foreign leaders were having no problems getting through to him by phone.
And he denied that he had requested security clearances for his children, in order to recruit them as advisers.
He did not address whether such clearance had been sought for Mr Kushner, his son-in-law, as reported by NBC News.
Mr Trump is due to be inaugurated as president on 20 January 2017.
Donald Trump has spent the week since his election holed up in the Manhattan skyscraper that bears his name.
He has been looking to fill key posts in his cabinet but the early indications are it is 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.
"In a normal transition to a normal administration, there's always disorder," writes senior Republican national security expert and Trump opponent Eliot Cohen in the Washington Post.
"This time may be different... The president-elect is surrounding himself with mediocrities whose chief qualification seems to be unquestioning loyalty. By all accounts, his ignorance, and that of his entourage, about the executive branch is fathomless."
"Every administration tends to have ideological divisions, to rely on an old guard of party people alongside its newcomers, innovators and ideological insurgents, Ross Douthat writes in the New York Times.
"But in this case, apart from the infamous-but-still-marginal alt-right and the small clutch of conservative intellectuals for Trump, there is really no Trumpist new guard at all, at least among the people qualified to staff a presidential administration."
Fox News, however, characterises the recent sackings as part of Mike Pence's campaign to remove lobbyists. "The move to get rid of lobbyists in key roles was one of the first decisions made by Vice President-elect Mike Pence in his role overseeing the construction of a Trump administration," it said.
"One source said the decision to remove the lobbyists 'makes good on [Trump's] vision of how he wants his government constructed'."
Politico quoted a Trump insider as describing the transition battle as a "knife fight".
"Trump continues to build his administration as he did his corporation and then his drama-heavy but ultimately successful campaign: with warring factions that will guarantee that the constant chaos and palace intrigue will continue in the White House," the website said.
"The idea of a 'team of rivals' isn't exactly a novel approach for a president filling out a cabinet. But this one may prove to be less of a team than a viper pit."
US President-elect Donald Trump has defended his handling of the transition to the White House, amid reports of disarray in his team.