Democratic attorneys general from states including Washington, New York and California launched the lawsuit.
The legal action objects to the policy of refusing entry to asylum seekers at the US-Mexico border.
Meanwhile, Vice-President Mike Pence has warned undocumented immigrants not to "risk the lives of your children" by trying to enter the US illegally.
At a news conference in Brazil, Mr Pence said he had a message "straight from my heart" for those planning a journey to America: "If you can't come legally, don't come at all."