Karzai's spokesman Aimal Faizi has said the letter would be tantamount to an apology, though not directly using that word, a report in the New York Times said.
In return for a letter from Obama, that will be a display of contrition by the President for military mistakes that have hurt Afghans, Karzai would end his strong opposition to American counterterrorism raids on private Afghan homes, an issue that has become very contentious between the two allies.
Once the conditions are met, the way would be cleared for an agreement to keep a smaller American troop force in the country after the 2014 drawdown.
Any final deal will have to have the approval of an Afghan grand council of elders or the 'loya jirga' which is to begin its meeting tomorrow.
According to Faizi, US Secretary of State John Kerry has offered to write the letter assuring the importance of an agreement and acknowledging American mistakes but Karzai wants the letter from Obama instead.
Faizi said Kerry has agreed to those terms, the NYT report added.
However, US National Security Adviser Susan Rice has flatly denied that there would be any presidential apology.
"No such letter has been drafted or delivered," Rice said in an interview to CNN.
"There is not a need for the United States to apologize to Afghanistan. Quite the contrary. We have sacrificed and supported them in their democratic progress and in tackling the insurgency and al Qaeda."
White House spokesman Jay Carney also did not confirm whether a letter will go from Washington to Kabul saying that the negotiations are "ongoing" and any agreement would not be final until it goes through the loya jirga.
A senior State Department official, speaking on condition of anonymity, said a letter acknowledging past issues like civilian casualties was a possibility being weighed.
"We will consider his request for reassurances, including the option of a letter from the administration stating our position," the official said in the NYT report.
In return for the letter, Karzai would accept wording in the troops deal that will allow American Special Operations raids to search and detain militants within Afghan homes, but only under "extraordinary circumstances" to save the lives of American soldiers.