North Korea's state-run news agency, KCNA, said Friday that U.S. special envoy for North Korea Stephen Bosworth gave the letter to North Korea's First Vice Foreign Minister Kang Sok Ju on December 9.
The U.S. State Department said earlier this week that Bosworth had taken Mr. Obama's letter to Pyongyang. Neither side revealed the contents of the letter.
But South Korea's Yonhap news agency says Mr. Obama has proposed setting up a liaison office in North Korea if the communist country returns to six-party nuclear disarmament talks.
A U.S. liaison office in Pyongyang would represent the first ever permanent and direct diplomatic interaction between the two countries.
Analysts say such an office would give the United States and North Korea an opportunity for regular dialogue on practical issues, such as a peace treaty that would formally end the 1950s Korean War, as well as ending the North's nuclear weapons programs.
After returning from Pyongyang, Bosworth said that six-nation talks on ending the North's nuclear programs are likely to resume soon, but that no date has been set.
Some information for this report was provided by AP.