The source said that Andonovski had handed in his resignation after he and the federation agreed to a mutual parting of ways.
The U.S. were eliminated in the last 16 by Sweden after a penalty shoot, the earliest the team has ever departed from a World Cup.
The Americans had won the previous two World Cups, in 2015 and 2019 with Jill Ellis in charge of the team.
Andonovski, an American citizen who was born in North Macedonia, took over the team in October 2019 having made his name in the National Women's Soccer League.
He is likely to have opportunities to return to the club game in America and could also be tempted to move abroad.
Under Andonovski the USA won a bronze medal in the Tokyo Olympics after losing to eventual champions Canada in the semi-finals.
The Athletic reported that Twila Kilgore, assistant coach of the team, was expected to take over as interim head coach.
A spokesperson for the U.S. Soccer Federation did not respond to a request for comment.
The USA headed to the World Cup amidst hype at home that suggested they would become the first team, male or female, to win three straight World Cup titles.
But after an unconvincing 3-0 defeat of Vietnam in their opening game of their group, they drew 1-1 with the Netherlands before barely managing to make it to the knockout stage with a 0-0 draw with Portugal.
The team's dancing celebrations after that goalless draw drew criticism from some pundits, including former captain Carli Lloyd.
But while they performed better against Sweden they couldn't find the target in 120 minutes and exited after the shoot-out.
The next coach is likely to lead a change of generation with 38-year-old Megan Rapinoe retiring and 34-year-old Alex Morgan and 35-year-old Kelly O'Hara entering the latter stages of their careers with the team.
While there is plenty of young talent expected to be ready for the 2027 World Cup, the immediate task will be to get the team in shape for the Paris Olympics next year.