In the worst-case scenario, according to State Department officials, an estimated 500,000 people could be killed while more than a million could be rendered homeless if the dam, Iraq’s largest, were to collapse in the spring, when the Tigris is swollen by rain and melting snow. The casualty toll and damage would be much less if Iraqi citizens received adequate warning, if the dam collapsed only partially or if it were breached in the summer or fall, when the water level is lower.

President Obama underscored the need to make emergency repairs in a call Wednesday with Prime Minister Haider al-Abadi, in which the two also discussed recent advances by Iraqi security forces against the Islamic State in Ramadi, in western Iraq, the White House said. Gen. Joseph F. Dunford Jr., the chairman of the Joint Chiefs of Staff, reiterated the urgent need to maintain the dam during a meeting with Mr. Abadi in Baghdad on Thursday, according to a senior Defense Department official.

Worried that repairs might not come in time, American officials have urged the Iraqi government to begin warning its citizens, including those who live under the control of the Islamic State in Mosul, about precautions to take and where to flee if the dam starts to fail.

A State Department official who is part of an interagency team assembled last year to focus on the problem cautioned that spring brings heightened risk. The official said the administration’s assessment was that absent energetic efforts to repair the dam, it would collapse — although he did not predict when.