Trump held meetings at his golf resort in Bedminster, New Jersey, with candidates for senior administration jobs after he takes office on Jan 20. On Saturday, he conferred with Mattis and Mitt Romney, formerly a fierce Trump critic now under consideration for secretary of state.
Christie, the Republican governor of New Jersey, advised Trump during the presidential campaign but was dismissed as the head of his transition team.
Asked by reporters before the meeting whether there was a place for Christie in his administration, Trump sidestepped the question, but called him "a very talented man, great guy."
Giuliani, the former New York mayor, was a candidate for secretary of state "and other things," Trump said.
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Trump met with Wilbur Ross, the billionaire investor, who he said was under consideration for commerce secretary.
Asked whether he wanted the job, Ross told reporters: "Well, time will tell." Trump also met with Kansas Secretary of State Kris Kobach, who is known for his hardline position on illegal immigration.
If selected, Mattis would be the first former flag officer, the very highest ranks of the military, to become US defense secretary since George Marshall took the job in 1950.
Trump wrote on Twitter on Sunday that "General James 'Mad Dog' Mattis, who is being considered for Secretary of Defense, was very impressive yesterday. A true General's General!"
From 2010 to 2013, Mattis headed the US military's Central Command, which oversees operations stretching from the Horn of Africa through the Middle East and into Central Asia including Afghanistan and Pakistan.
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During that time, he was at odds with the Obama administration on the need to prepare for potential threats from Iran and about resources for Afghanistan.
Mattis, 66, served as an American commander in the wars in Iraq and Afghanistan and was known to be popular among the troops.
Speaking on "Fox News Sunday," Vice President-elect Mike Pence, who now heads Trump's transition team," said Mattis had "a legendary military career."
Incoming White House Chief of Staff Reince Priebus said it was a "very real possibility" Mattis would get the job, telling the ABC program "This Week," "I know that President-elect Trump loves leaders like General Mattis."
Romney under 'serious consideration'
Pence said Trump and Romney had a good meeting and "a warm and a substantive exchange."
"I can say that Governor Romney is under active and serious consideration to serve as secretary of state of the United States," Pence said on the CBS program "Face the Nation."
Romney, the unsuccessful 2012 Republican presidential nominee, was a leader of the Republican establishment movement that tried to block Trump from becoming the nominee this year. In March, Romney called Trump "a phony," "a fraud" and "a con man."
A source close to Romney from his time as Massachusetts governor expressed concern he might be "frozen out" by officials whose thinking appears to be closer to Trump's, such as Michael Flynn, the president-elect's choice for national security adviser, Mattis, White House counselor Steve Bannon, and members of Trump's family.
"How much influence and latitude he will have will be up to Trump, and they don't appear to be on the same page about much," the source said.
Former Texas Governor Rick Perry, who made an unsuccessful bid for the 2012 and 2016 Republican presidential nomination, will meet with Trump on Monday and is being considered for Cabinet posts including defense, energy and veterans affairs, Trump's transition team said.
Conservative radio talk show host Laura Ingraham confirmed on Sunday she is being considered by Trump to serve as White House press secretary.
Democrats send signal of cooperation
When Trump takes office in January, both chambers of Congress will be controlled by his fellow Republicans. He could, however, face fierce opposition from Democrats to many of his legislative initiatives and some of his appointments.
But incoming Senate Democratic leader Charles Schumer, who hails from Trump's home state of New York, held out the possibility that Democrats could work with Trump on legislation to increase spending on US infrastructure.
Schumer said on "Fox News Sunday" he had talked to Trump about infrastructure and said it was possible that "we could get a major infrastructure bill done," maybe even in the first 100 days of his presidency.
"It has to be large and bold. Trump has talked about a trillion dollars. Good," Schumer said.
Trump said on Twitter that "I have always had a good relationship with Chuck Schumer" and the senator "has the ability to get things done."
Priebus offered mixed signals on Sunday on whether Trump would create a registry for Muslims in the United States, first saying: "I'm not going to rule out anything."
But he then said: "We're not going to have a registry based on religion" but there are "some people that are radicalised" and "some people who have to be prevented from coming into this country."
Trump appears to be setting up his administration to take a hard line confronting Islamist militancy.
Regarding Islam, Priebus said on "This Week" that "clearly there are some aspects of that faith that are problematic. And we know them. We've seen it. But it certainly isn't a blanket for all people of that faith."
Trump said he planned to live in the White House but that his wife, Melania Trump, and the couple's 10-year-old son, Barron, would not move in immediately. He said they would move to the White House "very soon, right after he finishes school."
Jason Miller, communications director of Trump's transition team, said earlier there was "some sensitivity" to moving the boy out of his current New York school in the middle of a school year.
