The State Department declined to comment on the report.
She'll also take up the administration's efforts to defend Israel at the United Nations and counter what Haley called the organization's bias against the Jewish state.
She has served under two secretaries of state - Rex Tillerson and her current boss Mike Pompeo - and was designated acting under-secretary for public diplomacy and public affairs from March to October 2018. Officials at the White House at times viewed Haley with skepticism, in part over her willingness to publicly disagree with the president.
Nauert, 48, is an unorthodox 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.
The nomination was first reported by Bloomberg News.
Ms Nauert is described by USA media as an "unorthodox choice" for the role due to her little experience in government or foreign policy.
Donald Trump urges 'full and complete' jail term for Michael Cohen
The tweet about Stone drew immediate criticism from several lawyers, who said it amounted to witness tampering . Earlier, he fired at his former personal attorney Michael Cohen for taking a plea deal in the case.
Tottenham midfielder Dier: Tough losing - but at least Arsenal competitive
The Spaniard's side will next travel to Old Trafford to take on Manchester United on Wednesday and will hope to keep their 12-game unbeaten run in the Premier League intact.
Trump says US-China ties make ‘BIG leap forward’
He also said that he was open to ratifying the previously unapproved Qualcomm-NXP deal should it be presented to him once again. In return, China is to purchase "very substantial" amount of agricultural, energy, industrial and other product from the US.
Two administration officials confirmed Trump's plans.
Haley rattled the United Nations when she arrived in January 2017 vowing that the United States will be "taking names" of countries that oppose Trump's foreign policy, particularly regarding Israel.
Trump had reportedly also been considering ex-White House aide Dina Powell, US Ambassador to Germany Ric Grenell, and US Canada Ambassador Kelly Croft for the job. With a master's degree from Columbia University's Graduate School of Journalism, she had come to Fox from ABC News, where she was a general assignment reporter.A graduate of Keith Country Day School in Rockford, Nauert is the daughter of the late Peter W. Nauert who led Pioneer Life Insurance Co. of IL.
Haley took a particularly forceful posture against Russian Federation, assailing the Kremlin at United Nations security council meetings over its aggression in Ukraine and support for Bashar al-Assad in Syria, even as Trump remained reluctant to criticize Vladimir Putin.