Trump and Vice President-elect Mike Pence have met with about 70 people so far as they look to shape their White House and Cabinet team before taking office on Jan 20. Pence told reporters there would be "a number of very important announcements" on Tuesday.
Republican US Representative Tom Price of Georgia, an orthopaedic surgeon, could be named health and human services secretary as early as Tuesday, the Atlanta Journal-Constitution reported, citing a dozen unnamed Republican sources.
A source close to the transition team said former Democratic US Representative Harold Ford of Tennessee was in contention to be transportation secretary. Politico, citing a source close to the transition, reported that Republican US Representative Lou Barletta of Pennsylvania was also being considered for that job. The Cabinet posts are subject to Senate confirmation.
Trump's consideration of Petraeus, who has also been mentioned as a contender for the top job at the Pentagon - adds a new layer of drama to his unusually public deliberations over the top diplomatic job for his administration.
"Just met with General Petraeus--was very impressed!" Trump said on Twitter shortly after Petraeus, a retired general, left an hour-long meeting with the Republican winner of the Nov 8 election at Trump Tower in Manhattan.
Petraeus, who led international forces in both Iraq and Afghanistan, was sentenced to two years' probation and fined $100,000 last year for the unauthorised removal and retention of classified information.
He admitted sharing classified information with his biographer, with whom he was having an affair. The scandal forced Petraeus to resign from the CIA in 2012.
Petraeus said after meeting Trump that the New York businessman "basically walked us around the world" in their discussion. "He showed a great grasp of the variety of challenges that are out there and some of the opportunities as well," Petraeus told reporters.
On Tuesday, Trump plans to hold a second meeting with Mitt Romney, the 2012 Republican presidential nominee, and will also meet with US Senator Bob Corker of Tennessee, the chairman of the Senate Foreign Relations Committee. Both men are in the running for the post.
A Republican source close to the transition team said it had appeared late last week that Trump was leaning toward choosing Romney as his secretary of state but that the appearance of Petraeus at Trump Tower suggested the president-elect was still undecided and casting a wider net for the position.
Former New York Mayor Rudy Giuliani, favoured by Trump loyalists who worked on the election campaign, remains in the mix for the job of America's top diplomat, Trump aides say.
Frances Townsend, a national security aide during the administration of Republican President George W Bush, also met Trump on Monday.
Petraeus troubles
Petraeus' past mishandling of classified documents is unlikely to be an obstacle to Trump offering him a top government post, even though Trump harshly criticised Democratic rival Hillary Clinton during the campaign for using a private email server while she was secretary of state.
Trump often compared the prosecution of Petraeus with the lack of legal action against Clinton, who was investigated by the Federal Bureau of Investigation but never charged with any offense.
"Just based on his public statements, I think (Trump) sees Petraeus as a good man who made a mistake, who did a fraction of what other people have done and received a lot more punishment," said a source who has advised the transition team on national security.
Among the Trump transition team, Petraeus and Romney are supported by Republicans looking for a candidate with gravitas who can unify the party, the source said.
But Kellyanne Conway, his campaign manager, has said Romney would cause a backlash among his supporters, who back Guiliani.
Ideological conservatives hope Trump picks John Bolton, the former ambassador to the United Nations, the source said. Corker is well regarded, but some in the Trump camp do not want too many senators coming into the Cabinet, the source said.
Trump arrived back in New York on Sunday after several days in Florida for Thanksgiving. He caused a stir on Sunday when he alleged on Twitter that “serious voter fraud” had occurred in the presidential election in New Hampshire, Virginia, and California - states won by Clinton - but provided no evidence to back his assertion.
The White House said on Monday there had been no evidence of widespread election fraud in the presidential contest, and all three states rejected Trump's allegation.
