One of the reports came from Fox News, where Ms Nauert has previously worked as a presenter.
Nikki Haley had announced in October that she was leaving the post by the end of the year.
Ms Nauert, 48, was made state department spokeswoman in April 2017, her first government position.
Fox News said Mr Trump would tweet about the appointment on Friday morning, while Bloomberg said it had been told of the decision to appoint Ms Nauert by three sources.
The state department has made no comment.
Bloomberg said Ms Nauert would be an unorthodox choice, given the little experience she had in government or foreign policy before she took up the role at the state department.
In addition to that role, Ms Nauert was appointed acting undersecretary for public diplomacy and public affairs in March this year.
Her appointment to the UN role would have to be approved by the US Senate.
Ms Nauert worked for Fox News from 1998 to 2005 and, after two years away during which she worked for ABC, she returned to Fox in 2007, later becoming a presenter for Fox & Friends.
Fox News has been a consistent supporter of the president and he often cites its news.
Ms Haley has served nearly two years in the post.
The 46-year-old former South Carolina governor - one of the few women in the Trump cabinet - gave no reason for leaving, but dismissed speculation she was planning to run for president in 2020.
President Trump said she had done "an incredible job".
Former deputy security council adviser Dina Powell and Mr Trump's daughter Ivanka were names initially floated as possible replacements.