Nauert was criticised last month, when she posted a tourist-style Instagram photo from outside a royal court in Riyadh; she was in Saudi Arabia on an official visit with Pompeo to discuss the fallout from the murder of journalist Jamal Khashoggi.
If she gets the job, Nauert would take the post with less clout than Haley, a former SC governor who announced in October that she would step down at the end of this year.
During her tenure as a State Department spokeswoman, Nauert largely stuck to the department's official script, often deflecting questions about U.S. allies, including Israel. If her much more recent experience at the State Department provides such rich fodder for critics, it portends a hard confirmation process ahead.
Nauert, 48, is an unusual choice for the United Nations role given that she had little experience in government or foreign policy before joining the administration in April 2017 after several years as an anchor and correspondent for Fox News, including on the "Fox and Friends" show watched by Trump.
Under Tillerson, Nauert grew frustrated at not having close access to her immediate boss and had been seen as quitting to return to NY, where her family has remained.
However Ms Haley has made no secret of the fact that she has had policy disagreements with Mr Trump.
Is Nauert up for the new role?
"If I were the State Department's spokesperson, on an official trip to learn more about the dark, dubious circumstances surrounding the disappearance of a U.S. resident, I probably would have skipped the tourist-in-Riyadh photos".
She was in charge of the Global Engagement Centre that fights extremist messaging from the Islamic State group and others, and she has a seat on the US Agency for Global Media that oversees government broadcast networks such as Voice of America.
He said USA influence is at a "low ebb", partly because of the absence of empowered ambassadors under Trump. "For Nauert it is going to be more about implementing than developing policies".
Without elaborating on the topics, she added, "For the most part he's been very willing to listen and very willing to come around". And the pick for the high-profile U.S. policy megaphone, seems more than justified, at least at first glance.
"Truly no one believes me: I am not even thinking about it", Haley said. She was an anchor for Fox and Friends, the Fox News Channel morning program that Trump says he watches regularly.
Nauert will arrive at the United Nations at a time when the Trump's administration's attitude to the organisation, always sceptical, is turning disdainful following the arrival of Bolton in the White House in March.