BEIRUT—Spiraling violence and advances by al Qaeda-linked fighters in Iraq, Syria and Lebanon are underscoring the cost of Syria's civil war as it increasingly spills over the country's borders.

The rise of the Islamist forces in Iraq is particularly worrisome to the Obama administration. In response, U.S. officials said Sunday they were seeking to boost military support—though they emphasized no troops—for Iraqi Prime Minister Nouri al-Maliki to help in his campaign to push back al Qaeda. U.S. officials are also considering...