If named, she would take over from outgoing ambassador Nikki Haley who announced her resignation last month without offering details on why.
Numerous reports in the U.S. media have named Nauert as a frontrunner for the post, citing sources close to the matter.
Asked to confirm the move during a question and answer session with reporters, Trump said: "She's under very serious consideration.
"She's excellent, she's been with us a long time, she's been a supporter for a long time, really -- excellent. So she's under very serious -- we'll probably make a decision next week."
Nauert was an anchor and correspondent for conservative cable station Fox News before joining the State Department in April 2017.
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.
Other possible contenders include Ric Grenell, the US ambassador to Germany, Jamie McCourt, the US envoy in Paris, and Kelly Craft, the ambassador in Canada, according to CNN.