WASHINGTON — The president of Yemen, Ali Abdullah Saleh, has long been known as a devious and unpredictable ruler, famous for his brinkmanship.

Yet in recent days, as Yemen slips from political crisis into bloody civil conflict, Mr. Saleh’s evasions have become downright bizarre — exasperating kings, presidents, and even many of his own loyalists. Three times, he has promised to sign an agreement to transfer power in the face of vast street protests, only to back out at the last minute for reasons that seem trivial.

“Even by his own standards of what is rational, he is not being rational,” said one American official, who spoke on the condition of anonymity under diplomatic protocol.

Last week, John O. Brennan, the White House counterterrorism chief, called Mr. Saleh and told him that if he signed the agreement to step down, President Obama would single out the Yemeni president as a positive example of change in his long-awaited speech on the Middle East, according to another administration official. The United States and Saudi Arabia have grown deeply worried about the political vacuum in Yemen, which has been a boon to Al Qaeda’s local affiliate.