The three royals’ message to Mr. Obama was a not-so-subtle slap at Qatar and Turkey—both of which, officials in these Arab countries believe, are funneling funds and possibly weapons to groups promoting political Islam and in particular to those aligned with the Muslim Brotherhood.

They are also concerned that aid from Qatar has bolstered the Al-Nusra Front, a powerful Syrian militia fighting Mr. Assad’s forces, which the U.S. has designated as a terrorist organization.

“We need someone to manage the players” in the region, said a senior Arab official involved in the discussions. “The U.S. and the president are the only ones who can put Qatar in its place.”…

Arab officials said they haven’t pressed Mr. Obama to deploy American troops in Syria or to use U.S. warplanes. Instead, they hope he will play a higher-profile role in seeking to forge a moderate, unified coalition that is purged of “radical” elements that threaten Syrian minorities who might otherwise already have broken with Mr. Assad.