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".
"As president, I cannot allow people into our country who want to do us harm," he added.
Mr Trump has already said the ban would take effect within 72 hours of court approval.
What does the ruling say?
The Supreme Court said in Monday's decision: "In practical terms, this means that [the executive order] may not be enforced against foreign nationals who have a credible claim of a bona fide relationship with a person or entity in the United States.
"All other foreign nationals are subject to the provisions of [the executive order]."
The ruling also said it would permit a 120-day ban on all refugees entering the US to go into effect, allowing the government to bar entry to refugee claimants who do not have any "bona fide relationship" with an American individual or entity.