Trump is scheduled to visit the ornate, marble-clad courthouse in Washington for the investiture of new Justice Neil Gorsuch, whose Senate confirmation in April was his first major accomplishment.
Attention will be focused on whether Trump, known for his off-the-cuff remarks and incendiary tweets, will follow the rules of an institution known for its courtesy and tradition.
The stakes are heightened by the fact that Trump's so-called travel ban, one of his signature policies, is now before the justices after being blocked by lower courts.
The president is expected to sit in the courtroom during the brief ceremony in which Chief Justice John Roberts will administer the judicial oath to Gorsuch.
Trump is not expected to make a speech at the event, but he is likely to talk briefly to the justices beforehand in the court's conference room, as other presidents have done in the past, according to a court spokeswoman.
In deciding whether to allow the travel ban to go into effect, the justices are set to weigh whether Trump's harsh election campaign rhetoric can be used as evidence that the March 6 order was intended to discriminate against Muslims.
Trump has spoken out against courts blocking the ban and has also criticized his own lawyers. The court is currently considering an emergency request from the administration seeking to put its travel ban into effect while litigation continues.
Federal judges in Maryland and Hawaii blocked Trump's 90-day ban on travelers from Libya, Iran, Somalia, Sudan, Syria and Yemen. The Hawaii judge also blocked a 120-day ban on refugees entering the United States. The injunctions blocking the ban were upheld on appeal.
Trump's appointment of conservative Gorsuch has been his most significant win since taking office in January.
Gorsuch, who has been sitting on the bench since April 10, restored the high court's 5-4 conservative majority. There was a vacancy on the court for more than a year following the death of conservative Justice Antonin Scalia.
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