The Republican sought to laugh off the comment, which was preceded by the supporter saying: "We have a problem in this country - it's called Muslims".
The comments were made at a campaign rally for Mr. Trump in New Hampshire.
Democratic frontrunner Hillary Clinton said his failure to denounce "hateful rhetoric" was "disturbing and wrong".
Pointing to his first questioner at the campaign event in Rochester, the billionaire businessman said: "I like this guy."
"We have a problem in this country called Muslims," the man said. "We know our current president is one. You know he's not even an American."
"We need this question?" Mr Trump said, laughing.
"But anyway," the man continued, "we have training camps growing where they want to kill us. That's my question: When can we get rid of them?"
Mr Trump failed to clarify that Mr Obama is a Christian American, instead replying that "bad things are happening" and saying he would look into them.
Mrs Clinton joined criticism of Mr Trump on social media, tweeting: "Donald Trump not denouncing false statements about POTUS & hateful rhetoric about Muslims is disturbing & just plain wrong. Cut it out."
In a statement, Democratic National Committee chair Debbie Wasserman Schultz said Mr Trump's "racism knows no bounds".
