Painting by graffiti artist was among official gifts to Obama from David Cameron on his first trip as PM
The work, Twenty First Century City, is by Ben Eine, said to be one of the PM's wife Samantha's favourite artists.
Obama continued the art theme by presenting the Camerons with a signed lithograph by pop artist Ed Ruscha.
The piece, Column with Speed Lines, was chosen for its red, white and blue colours matching the British and American flags.
Famous artist
Writing on his website, Ben Eine said it had been a weird day because "David Cameron has given one of my paintings to President Obama in an art swap".
The artist, who has worked with Banksy, began his career by 'tagging' his name on buildings and trains around east London, examples of which are featured on his website under the heading 'vandalism'.
In recent years he has been asked to decorate shutters in cities as far afield as Tokyo and New York.
His works, many dominated by the strong lines and empty streetscapes of Southern California, command large sums.
Other gifts from the Camerons to the Obamas included two pairs of Hunter Wellington boots, one in pink and one in purple, for the first family's two daughters.
For Michelle Obama, there were candles made by London-based perfumer Miller Harris.
David Cameron and Barack Obama swapped jokes at a White House news conference.
Cameron's six-year-old daughter Nancy was given a silver charm necklace, and his son Elwen(4), was given a
DC United football shirt.
Samantha Cameron, who is expecting a child in September, received a gift basket including a baby blanket.
Beer bet
When Cameron met President Obama at the G8/G20 summit in Toronto last month, they exchanged bottles of beer, following a bet on England's World Cup clash with the US, which ended in a 1-1 draw.
Classic DVDs
The US president presented Cameron with a bottle of Goose Island 312 from his home city of Chicago, while Cameron gave Obama a bottle of Hobgoblin, made by the Wychwood brewery in his Oxfordshire constituency.
Swapping notes on the beers at a White House news conference on Tuesday, Obama said the British bitter was excellent but confessed, "I did drink it cold."
Cameron joked that the American beer "was obviously very effective" because he had ended up cheering on Germany against Argentina while drinking it, "something that's a big admission for a British person."
Obama was criticised for lack of thought when he presented previous Prime Minister Gordon Brown with a set of 25 DVDs of classic American films, when Brown visited Washington.
Brown gave Obama a pen holder made from the timbers of a Victorian anti-slavery ship.
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