Moore called Damon's political stances in recent years courageous and urged him to run, despite there being no hint from the actor that he would care to.
In a nod to the acting past of Republican President Ronald Reagan, Moore said: "The Republicans have certainly shown the way that when you run someone who is popular, you win.
"Sometimes even when you run an actor, you win."
The suggestion quickly spread across the media, generating a lot of chuckles as well as predictable outrage from conservative pundits.
But the suggestion showed two things that are not so easily dismissed. First, quietly and with impressive charm, Damon has emerged as an eloquent and fierce spokesman for a slice of liberal America.
He has been happy to take a stand on everything from the Iraq war to education, and rather than praise the President, he says Obama has "mishandled his mandate".
Secondly, it showed that America, more than any country in the world, has a fluid boundary between the worlds of entertainment and politics.
From Reagan to Clint Eastwood, Sonny Bono to Arnold Schwarzenegger and Al Franken and many more, the list of United States actors and performers turned politicians is lengthy and even distinguished.
"The kind of character that pursues an acting career in America is often the same kind of character that pursues a political career. You have to stand up and make people like you and be good on TV," said Professor Robert Thompson, a popular culture expert at Syracuse University.
So, Matt Damon for President? In 2012, almost certainly not. But one day? You never know.
Damon is certainly no shallow celebrity, long on good looks but short on brains. The Massachusetts native may have chosen Hollywood as a career, but he is not an actor picking causes with carefully staged press conferences on subjects that no one could dislike, such as stopping African famine.
Instead, Damon has lent his high- profile name to the distinctly unfashionable cause of the Working Families Party. The WFP is an obscure leftwing political party that exists as a sort of pressure group in New York state on Democrats and leftists to pursue progressive ideals.
Attaching your name to the WFP is about as far from trendy as any Hollywood celebrity could get. Yet Damon has been a passionate advocate for the party, appearing last year in a campaign video for it in which he urged New Yorkers to shun the Democrats and vote for the WFP as a genuine leftwing alternative.
Damon has won the hearts of many liberals by criticising Obama over policy issues, and standing up for teachers. Speaking at a recent Save Our Schools march in Washington, he angrily rejected a reporter's suggestion that teachers were cosseted.
"A teacher wants to teach. Why else would you take a shitty salary, and really long hours, and do that job, unless you really love to do it?" he fumed.
A video of the encounter went viral, and Damon was hailed as a hero by teachers' groups.
Damon, like Sean Penn with Haiti and George Clooney with Darfur, is one of the few big names who can say they are activists, not just celebrity brands attached to a good cause.
He founded the H2O Africa Foundation, which later became Water.org and which aims to bring clean water to disadvantaged people. He has been involved with Darfur.
He has spoken eloquently and publicly on subjects dear to liberal hearts. He has slammed the recent debt-ceiling deal struck by Obama and the Republicans and called for rich people like himself to be taxed more. He opposed the Iraq war.
Perhaps one should not be surprised; Damon is highly educated. Though he eventually dropped out to pursue an acting career, he went to Harvard, where he studied English. His mother - who introduced him at the teachers' rally - is an education professor.
But, experts say, Hollywood has given him what is needed most: name recognition.
"An actor has a precious thing in politics. People know who they are and they will pay attention when someone puts a microphone in front of them," said Thompson.
That power can make a political career out of the unlikeliest of raw material. Look at how former pro wrestler Jesse Ventura became governor of Minnesota, or how comedian Franken became a senator from the same state and - perhaps most bizarre of all - how Schwarzenegger went from playing a killer robot to being governor of California.
But Franken, a former stalwart of Saturday Night Live, has won plaudits for his seriousness as a politician. Schwarzenegger was seen as a joke when first elected, but he easily won a second term and became known for cutting-edge environmental policies.
Most successful was Reagan, who went from a B-movie actor to being one of the most influential Republican Presidents of the 20th century.
But the road to political power is not always easy for an actor. On the liberal side of the aisle, Warren Beatty was mentioned as someone who might run for President but never did.
And among Republicans, the name of Fred Thompson stands as a salutary lesson in the limits of power. Few people have blurred the lines between acting and politics as much as Thompson, who combined his acting career with becoming a senator from Tennessee.
He has played a US President on TV but when he ran for the Republican nomination in real life in 2008, his attempt was a disaster.
So, while Americans are tolerant of actors who want to be politicians, they do not write them a blank cheque.
"Celebrity can be a blessing or a curse. You are able to get people to listen to you, but you need to have something they want to hear," said Robert Thompson.
From movie fame to high office
Sonny Bono: Best-known for his musical career in the duo Sonny and Cher, Bono became a Republican congressman and mayor of Palm Springs in California.
Fred Thompson: Had a long-running acting career, including playing the role of United States presidents on TV and a recurring role on Law & Order. He was also a Tennessee senator and ran unsuccessfully for the Republican presidential nomination in 2008.
Ronald Reagan: The most successful example of an actor turned politician. His road from Hollywood to the White House included terms as president of the Screen Actors' Guild and governor of California.
Arnold Schwarzenegger: The body-builder turned Terminator won two terms as Republican governor of California.
Clint Eastwood: The tough guy actor-turned-auteur director also dabbled in politics. He served one term as the mayor of Carmel, California.
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