State funerals, held in the US capital Washington DC, are steeped in tradition and usually only held for former presidents, the last being Gerald Ford in 2006.
The last non-president to be granted one was Gen Douglas MacArthur, the Second World War and Korean War soldier, in 1964, five years before Armstrong landed on the moon.
Such an event typically involves pall-bearers from five branches of the US Armed Forces, a series of artillery salutes, a fly-past and a number of bands and choirs.
The flag-draped coffin is taken in a horse-drawn gun-carriage and placed in the Capitol rotunda for a public viewing, and a service is held at the Washington National Cathedral.
Bill Johnson, a Republican congressman from Armstrong's home state of Ohio, led calls for the astronaut to get a similar honour.
"I ask President Obama to hold a state funeral for Neil Armstrong so that every American may pay tribute to this groundbreaking hero," he said. "His first step on the moon showed the world that Americans can do anything."
However, Mr Armstrong, a former US Navy test pilot, was a determinedly private figure who granted few interviews and shunned publicity. His family may object to a public funeral. A private service is expected to be held on Thursday or Friday.
As tributes flooded in for Armstrong, who died at a hospital in Ohio on Saturday aged 82, more details of his illness also emerged. In a statement his family said he suffered "complications resulting from cardiovascular procedures."
On August 6 he had failed a medical stress test and underwent quadruple bypass surgery the following day. He had four blockages in his coronary arteries.
A day after the open heart surgery fellow moonwalker Gene Cernan had relayed a message from Armstrong's wife Carol that he was "doing great," and that doctors expected there to be no problems with his recovery. He died less than three weeks later.
Following his death, fellow Apollo 11 astronaut Buzz Aldrin called his colleague the best pilot he had ever seen.
"I know I am joined by millions of others in mourning the passing of a true American hero and the best pilot I ever knew. I will miss my friend Neil," he said.
Astronaut Michael Collins, the third member of the mission who orbited the moon in the command module, said: "He was the best, and I will miss him terribly."
In a statement released on Saturday, his family paid tribute to "a reluctant American hero who always believed he was just doing his job."
"While we mourn the loss of a very good man, we also celebrate his remarkable life and hope that it serves as an example to young people around the world to work hard to make their dreams come true, to be willing to explore and push the limits, and to selflessly serve a cause greater than themselves.
"For those who may ask what they can do to honour Neil, we have a simple request. Honour his example of service, accomplishment and modesty, 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."
Other tributes poured in, led by President Obama who said: "Neil was among the greatest of American heroes – not just of his time, but of all time."
In May 2010 Armstrong had criticised Mr Obama's vision for the future of space travel, appearing before a Senate committee to say he believed the president had been "poorly advised."
Armstrong was unhappy with the failure to replace Nasa's defunct space shuttle programme.
It was a rare public appearance by the astronaut.
Armstrong's unlikely path to the moon began in Wapakoneta, Ohio, where his fascination with flying led him to work at a nearby airport when he was a teenager. He received his pilot's licence on his 16th birthday and went on to study aeronautical engineering.
As a US Naval Aviator, he flew 78 missions in the Korean War and later piloted more than 200 different models of pioneering aircraft as a test pilot.
He became an astronaut in 1962 and always modestly attributed his status as first man on the moon to chance.
"I wasn't chosen to be first," he said "I was just chosen to command that flight."
But Nasa had chosen him for good reason. When Eagle, the Apollo 11 lunar module, made its perilous descent to the surface of the moon on July 20, 1969, it was the extraordinary skill of Armstrong, then aged 38, that averted disaster.
As the tiny craft skated over lethal boulder fields and steep-walled craters, its fuel tanks emptying, Armstrong and fellow astronaut Buzz Aldrin realised they were miles from the planned landing spot.
Without breaking a sweat, Armstrong found a flattish patch of ground and put them down with only seconds of fuel left.
Armstrong also maintained to the end that his famous first words from the moon were misquoted.
Hundreds of millions of people on Earth heard him say "That's one small step for man, one giant leap for mankind," but the astronaut said he had spoken the words "a man."
Decades later, in 2006, a computer analysis found evidence that he was right, and the world was wrong. Nasa backed its hero, saying: "If Neil Armstrong says there was an 'a,' then as far as we're concerned, there was an 'a'."
xrefs to front of features and obits