President-elect Donald Trump pressed ahead Tuesday with efforts to build his cabinet with former New York mayor Rudy Giuliani tipped for secretary of state -- but reports said vicious infighting was hobbling the crucial process.

The Republican billionaire drew a barrage of criticism over his pick of chief strategist: the anti-establishment firebrand Steve Bannon, onetime head of the provocative Breitbart website seen by critics as a darling of white supremacists.

Top Trump ally Giuliani, hawkish former UN ambassador John Bolton, retired general Michael Flynn and Alabama Senator Jeff Sessions were all reported to be on the shortlist for a top administration job.

But the high-stakes process of filling more than a dozen cabinet posts has been tumultuous by many accounts. One source cited by CNN described the intense lobbying as a "knife fight."

Vice president-elect and transition leader Mike Pence spent much of the day at Trump Tower -- which has been a hive of activity since last Tuesday's vote -- but his only comment to media as he left was "Great day."

Among other sightings at the Manhattan high-rise was Ted Cruz -- the arch-conservative Texas senator who the real estate mogul attacked relentlessly during the Republican primaries, dubbing him "Lyin' Ted."

Asked whether he wished to be considered for a spot in Trump's administration or would remain in the Senate, he replied: "This election was a mandate for change."

"And I look forward to working hard to help lead the fight to actually accomplish the conservative agenda that Donald Trump and Mike Pence and Republicans across this country campaigned and promised the voters to deliver."

- 'Non-traditional names' -

Jason Miller, a transition communications adviser, told reporters Trump and Pence would be "reviewing a number of names" for cabinet positions, including "non-traditional names."

"People will be excited when they see the type of leaders the president-elect brings into this administration," he said.

But Trump's transition team has faced a string of setbacks as it tackles the daunting task of building an administration with the clout to support the 70-year-old political novice when he takes office in just nine weeks.

The first shake-up came Friday, when Trump reshuffled the team, placing Pence in charge. Then on Tuesday, the transition team's head of national security, Mike Rogers, resigned in what was interpreted as a new sign of disarray.

Further reinforcing the impression of tensions, The New York Times reported Tuesday that Trump had removed from the transition a second top defense and foreign policy official, consultant Matthew Freedman.

According to a US defense official, by mid-afternoon Tuesday the Pentagon transition team still had not been contacted by Trump's transition team.

- Giuliani for State, or Bolton? -

On Sunday, Trump named Reince Priebus, a mainstream Republican operative who backed Trump while chairman of the Republican National Committee, as his White House chief of staff.

Trump's choice of Priebus -- announced at the same time as Bannon -- suggested a leader torn between a promise to shake up Washington and the need to build a cabinet with political experience and connections with Congress.

According to a top Trump aide, Giuliani -- a member of Trump's inner circle -- is a "serious" contender to become the next secretary of state.

The crime-fighting former prosecutor was mayor of New York on 9/11, and his decisive leadership after the World Trade Center's twin towers were toppled in the September 2001 attacks made him a national hero.

But CNN reported that Team Trump was looking into whether the 72-year-old's business ties -- including work as a lobbyist for a Venezuelan oil firm -- could complicate his confirmation in the role.

Bolton, a neo-conservative hawk and former undersecretary of state, also was reported to be in the running for the top diplomatic post.

He was a controversial choice for UN envoy in 2005, having once said if the UN headquarters lost 10 floors, "it wouldn't make a bit of difference."

"John would be a very good choice," Giuliani said Monday at a forum sponsored by The Wall Street Journal.

Asked if there were anybody better, Giuliani quipped: "Maybe me, I don't know."

Giuliani outlined his foreign policy vision at the forum, putting the fight against the Islamic State group atop his agenda, and arguing that Russia was not a military threat to America.

His comments chime well with Trump's promise to improve ties with Moscow -- and his call for the United States to place less emphasis on ousting Syrian President Bashar al-Assad, whose main backer is Russia, and more on fighting IS.

In an interview aired Tuesday, Assad said that Trump would be a "natural ally" if he fulfills his pledge to fight "terrorists."

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