Ivanka Trump
Among those first to be cited by cable news commentators was Ivanka Trump, the President's 36-year-old daughter, who serves as an unpaid advisor to him in the White House.
Trump told reporters that he has heard his daughter's name discussed for the post. He says she'd be "incredible" in the role, but he knows if he selected her he'd be accused of nepotism.
"There's nothing to do with nepotism but the people that know, know that Ivanka would be dynamite," he said.
Ivanka Trump has spent most of her life in New York, and some speculated that she could be eager to return to the city.
Furthermore, the role would give her a high profile platform for her own policies - she has been strongly criticised for failing to enact, in public at least, her promises to champion womens' issues.
It would also put some distance between her and her father's contentious Administration in Washington.
However, it may be unlikely, given that her husband Jared Kushner is a senior adviser to the President - would he want to move back to New York, and lose his proximity? Would she and the children move, leaving him alone in DC?
Perhaps more significantly, Trump could struggle to convince Senators - who must confirm the appointment of the UN ambassador - that he was not breaking the nepotism law.
Ivanka Trump and Kushner avoided that charge by taking unpaid roles.
Ivanka Trump reportedly laughed today, when she was asked by White House colleagues if she wanted the job.
Dina Powell
Dina Powell resigned from the White House in January, stepping down after a year as deputy national security adviser for strategy.
The 45-year-old former Goldman Sachs banker, who was born in Egypt, was well-liked by Kushner and Ivanka Trump but clashed with Trump's former advisor Steve Bannon.
She is considered a reliable and professional choice - but whether she would want to leave her lucrative new posting at Goldman Sachs, where she returned on leaving the White House, remains in doubt.
The President says Powell is under consideration.
Richard Grenell
Another name in the mix is Richard Grenell, the current US Ambassador to Germany.
Grenell, 52, was the spokesman for four US ambassadors to the UN, holding the job from 2001 until 2008. As such he worked for John Bolton, the current national security adviser, who is likely to be extremely influential in choosing the next ambassador.
Grenell, however, may be a controversial choice.
He has caused waves in Germany by offending his hosts within hours of being sworn in, when he tweeted that German companies should immediately cease doing business with Iran. He then sparked anger by giving an interview to Breitbart, in which he said he was hoping to support nationalist movements, like the "America First" of Trump, throughout Europe - comments seized upon as showing support for far-Right groups in Germany.
"Don't put words in my mouth," he tweeted in response. "The idea that I'd endorse candidates/parties is ridiculous. I stand by my comments that we are experiencing an awakening from the silent majority — those who reject the elites & their bubble. Led by Trump."
Heather Nauert
Some suspect that Trump will want to replace Haley with another woman, given how few women currently serve in his cabinet (Linda McMahon, for small businesses; Kirstjen Nielsen, homeland security; Elaine Chao, for transport; and Betsy DeVos, education).
Nauert, a former ABC News correspondent and Fox News host, is currently spokesman for the state department.
The 48-year-old is seen as knowledgeable and cool-headed, and is already up-to-speed on US foreign policy issues.
Jon Huntsman
Jon Huntsman, a 2012 presidential candidate, is currently serving as the US Ambassador to Russia.
The 58-year-old former Utah Governor is highly respected: he has served in every presidential administration since Ronald Reagan, and was ambassador to Singapore under George W. Bush, before Barack Obama made him Ambassador to China.
His insight into Russia may be considered an asset.
Yet it remains unclear whether Trump would want such an important figure brought back from Moscow.
