The US Supreme Court has allowed the Trump administration to fully enforce a ban on travel to the United States by residents of six mostly Muslim countries.

The justices, with two dissenting votes, said Monday that the policy could take full effect even as legal challenges against it make their way through the courts.

The action suggests the high court could uphold the latest version of the ban that US President Donald Trump announced in September.

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The ban applies to travellers from Chad, Iran, Libya, Somalia, Syria and Yemen.

Lower courts had said people from those nations with a claim of a "bona fide" relationship with someone in the United States could not be kept out of the country.

Grandparents, cousins and other relatives were among those the courts said could not be excluded.

Justices Ruth Bader Ginsburg and Sonia Sotomayor would have left the lower court orders in place.

The San Francisco-based 9th US Circuit Court of Appeals and the 4th US Circuit Court of Appeals in Richmond, Virginia, will be holding arguments on the legality of the ban this week.

Both courts are dealing with the issue on an accelerated basis, and the Supreme Court noted it expected those courts to reach decisions "with appropriate dispatch".

Quick resolution by appellate courts would allow the Supreme Court to hear and decide the issue this term, by the end of June.