- 22 September 2016
- From the section US & Canada
Mr Obama said the letter was from a child "who hasn't learned to be cynical, or suspicious, or fearful".
The video has been shared more than 60,000 times on Facebook.
"Dear President Obama, remember the boy who was picked up by the ambulance in Syria?" Alex wrote in a letter published by the White House.
"Can you please go get him and bring him to our home ... we'll be waiting for you guys with flags flowers and balloons. We will give him a family and he will be our brother."
"We should all be more like Alex," the President wrote.
"Imagine what the world would look like if we were. Imagine the suffering we could ease and the lives we could save."
On social media, many users applauded the president's approach - but reserved most of their praise for young Alex.
"A six year old who has more humanity, love, and understanding than most adults. Kudos to his parents," a Texas woman commented on Facebook.
"I heard this earlier today, as read by my president," said another. "Even with that pre-conditioning, made me cry while reading it just now.
"Neither of these sweet little boys, someone's sons, are Skittles," she said - referencing a controversial Donald Trump campaign advert likening Syrian refugees to the popular brand of sweets.
The US President has urged developed nations to do more to help refugees from the Syrian crisis. In August, the White House said it had admitted 10,000 Syrian refugees to the United States this year.
It plans to to take another 110,000 in the 2017 financial year, Mr Obama said.