Mr. Obama is committed to ending America’s military involvement in Afghanistan by the end of 2014, and Obama administration officials have been negotiating with Afghan officials about leaving a small “residual force” behind. But his relationship with Mr. Karzai has been slowly unraveling, and reached a new low after an effort last month by the United States to begin peace talks with the Taliban in Qatar.

Mr. Karzai promptly repudiated the talks and ended negotiations with the United States over the long-term security deal that is needed to keep American forces in Afghanistan after 2014.

A videoconference between Mr. Obama and Mr. Karzai designed to defuse the tensions ended badly, according to both American and Afghan officials with knowledge of it. Mr. Karzai, according to those sources, accused the United States of trying to negotiate a separate peace with both the Taliban and their backers in Pakistan, leaving Afghanistan’s fragile government exposed to its enemies. …

The option of leaving no troops in Afghanistan after 2014 was gaining momentum before the June 27 video conference, according to the officials. But since then, the idea of a complete military exit similar to the American military pullout from Iraq has gone from being considered the worst-case scenario — and a useful negotiating tool with Mr. Karzai — to an alternative under serious consideration in Washington and Kabul.