SANA, Yemen — Houthi rebel militiamen seized control of the palace of Yemen’s president and clashed with guards outside his residence on Tuesday in an escalation of the violent crisis that has gripped the capital for days, raising fears of a coup in one of the Arab world’s most impoverished and insecure states.

The president, Abdu Rabbu Mansour Hadi, viewed by the United States as a crucial counterterrorism ally, was believed to be in the capital, but his exact whereabouts was unknown. He made no public statements as the fighting escalated, though Houthi leaders insisted that he was safe and in his home.

Later on Tuesday, the most senior Houthi leader, Abdel Malik al-Houthi, gave a televised speech indicating that the advances by his fighters were a warning to Mr. Hadi to accelerate political changes they have demanded, and not an attempt to depose him. But if the president does not respond, Mr. Houthi said, “all necessary measures will be open.”

The mayhem that has convulsed Yemen, which left at least eight people dead on Monday in Sana, has also left citizens facing a leadership vacuum as the country is seized by crises, including spreading armed conflict and widespread hunger. The turmoil has been increasingly worrisome to American officials because Yemen is the base of Al Qaeda in the Arabian Peninsula, which has asserted responsibility for a number of attacks, most notably the deadly assault on the Charlie Hebdo newspaper in Paris this month.