The Supreme Court said the order affects arguments scheduled for March 23-25 and March 30-April 1 because of "public health precautions."
"The court will examine the options for rescheduling those cases in due course in light of developing circumstances," it added.
Six of the nine justices are 65 and older, which puts them at higher risk over the coronavirus disease.
Oral arguments over Trump's federal and New York cases related to subpoenas over his financial records were scheduled for March 31. Both cases involve the question of presidential immunity regarding personal financial records. A ruling was expected by late June.
Other court business, such as a regular closed-door conference on Friday, may be held with justices remotely by phone. The court's release of orders on Monday will go on as normal.
The Supreme Court building is closed to the public and is working on expanding remote capabilities to reduce the number of employees present.
The postponement is the first time in over a century Supreme Court business has been interrupted. It last occurred during the 1918 Spanish flu epidemic, and the court also shortened its calendar in the late 18th century because of outbreaks of yellow fever.