America's highest court also granted a White House request allowing part of its refugee ban to go into effect.
The justices said they would consider in October whether the president's policy should be upheld or struck down.
Mr Trump seeks to place a 90-day ban on people from six mainly Muslim nations and a 120-day ban on refugees.
The president welcomed the ruling's qualified authorisation to bar visitors from Iran, Libya, Somalia, Sudan, Syria and Yemen, which he described as "terror-prone countries".
