Trump told reporters that Nauert, who is in line to take over from Nikki Haley, had done well at the State Department.
"She's very talented, very smart, very quick, and I think she's going to be respected by all," Trump said.
Nauert, 48, had been touted for the post since October when Haley, a former governor of South Carolina seen as entertaining future political ambitions, announced that she was stepping down.
Nauert -- a former anchor of "Fox and Friends," among the television-loving Trump's favorite shows -- became the spokeswoman of the State Department with no foreign policy experience.
Unlike Haley, she is not expected to have cabinet status, meaning that foreign policy decisions will remain firmly with Trump's hawkish national security advisor John Bolton, and Nauert's current boss, Secretary of State Mike Pompeo.
She will need confirmation by the Senate, where Trump's Republican Party enjoys a majority.
Early praise for the appointment came from Israel, which has long counted on the United States to veto unfriendly resolutions on the UN Security Council.
"Ms. Nauert has stood by the State of Israel in her previous positions, and I have no doubt that the cooperation between our two countries will continue to strengthen as ambassador to the UN," Israel's ambassador to the United Nations, Danny Danon, said in a statement.
Haley, in what could be a final diplomatic push, on Thursday failed in a bid for the UN General Assembly to condemn the Palestinian Islamist movement Hamas for firing rockets at Israel.
But for the United States, the vote also succeeded in reinforcing its stance that the world body is biased against Israel.