The Ohio native died Saturday in Cincinnati at age 82. No other information was released immediately about the service.
Obama on Monday issued a proclamation calling for U.S. flags to be lowered the day of Armstrong's burial, including at the White House, military posts and ships, U.S. embassies and other public buildings "as a mark of respect for the memory of Neil Armstrong."
Ohio Gov. John Kasich on Monday had Ohio flags on all public buildings and grounds flown at half-staff through Friday.
There have been preliminary discussions about a national memorial service for Armstrong, who often shunned publicity in the decades after his historic mission, but a family spokesman said there were no details yet.
U.S. Sen. Rob Portman, who is from Ohio and has called Armstrong "a good friend and adviser," will eulogize him at Friday's service.
Portman is in Tampa, Fla., for the Republican National Convention, where he is scheduled to speak Wednesday night. Spokeswoman Caitlin Dunn said his office is working on travel arrangements to get him back to Cincinnati in time for the service. The convention schedule has already been changed this week and could be further disrupted as Tropical Storm Isaac bears down on the Gulf Coast.
The Museum of Natural History & Science of the Cincinnati Museum Center has an exhibit that includes a moon rock and replicas of Armstrong's Apollo 11 spacesuit and tools used on the moon. It is offering free admission through Labor Day to honor Armstrong, and more than 2,000 people visited Sunday.
Armstrong, who commanded the Apollo 11 moon landing on July 20, 1969, was born in Wapakoneta, in western Ohio. He is celebrated there at the Armstrong Air & Space Museum, which is planning a memorial tribute Wednesday night.
The tribute is called "Wink at the Moon." The statement Armstrong's family released upon his death requested that the public honor his example of service, accomplishment and modesty, adding "and the next time you walk outside on a clear night and see the moon smiling down at you, think of Neil Armstrong and give him a wink."
The Armstrong family released another statement Monday evening saying that "the outpouring of condolences and kind wishes from around the world overwhelms us and we appreciate it more than words can express."
The statement suggested that instead of flowers, memorial contributions could be made to the Neil Armstrong New Frontiers Initiative at the Cincinnati Children's Hospital Medical Center or to the Neil Armstrong Scholarship Fund at the Telluride Foundation in Telluride, Colo.
Associated Press reporter Thomas Beaumont contributed in Tampa.