The prospect of an expanded alliance of world powers to combat Islamic State dimmed on Thursday as top U.S. and Russian leaders dug in on longstanding disagreements, pointing to vexing problems in the hopes for a “grand coalition” of nations.

Whether France, the U.S. and Russia can find common ground—particularly over their chief disagreement, the future of Syrian President Bashar al-Assad—will be at the crux of conversations next week when French President Francois Hollande travels to Washington to meet with President Barack Obama and then to Moscow to huddle with President Vladimir Putin.

For days, it has appeared the major powers would look for a way to resolve the question of Mr. Assad’s fate because of the urgency of last week’s attacks in Paris and the downing of a Russian passenger jet in October, for which Islamic State has claimed responsibility.

But Mr. Obama on Thursday said Russia and its ally, Iran, must decide if they will continue to back Mr. Assad or join the international effort to defeat extremists, laying down his marker at a summit in Manila, where he said the Syrian leader must leave office.

“At some point, the Russians and the Iranians are going to have to make a fundamental decision: Do they actually believe that they can prop up Assad and win on the ground militarily inside of Syria against all the opposition?” the president said. “Or do they actually think that it is better to save the Syrian state and work with the international community and the U.N. to find a government that truly can be legitimate?”

Hours later, Russian Foreign Minister Sergei Lavrov said the talk by Western leaders of a broad anti-Islamic State alliance showed that “levelheaded politicians are dropping secondary issues and realizing the need to concentrate on the biggest issue at hand: stopping ISIS’s attempts to spread influence globally.”

However, he said in an interview with Radio Russia that Mr. Assad “represents the interests of a significant part of Syrian society…so no peaceful solution can be found without his participation.”

Mr. Assad, meanwhile, said there would be no political process in Syria unless all those bearing arms surrender or are defeated. The regime considers all those fighting it to be terrorists, including opposition militants backed by the U.S.“You cannot achieve anything politically while you have the terrorists taking over many areas in Syria,” he said in an interview with Italy’s Rai Uno channel released on Thursday. Separately, he told French magazine Valeurs Actuelles he would never take orders from an outside power to leave office.

On Thursday, Messrs. Obama and Hollande spoke by phone, and French officials were in Washington preparing for the leaders’ meeting and emphasizing a sense of urgency about Islamic State.

“The crisis is deepening and destabilizing Europe.…We have to act quickly,” said a senior French official. “We need the continued strategic military pressure from the Americans.”

Mr. Hollande, like his U.S. counterpart, has long argued for the Syrian leader’s ouster, saying he can’t be part of a political solution in Syria. However, in the aftermath of the terrorist attacks, he said he wants to focus on the crisis presented by Islamic State.

Despite common cause on Islamic State, how the various countries and their Middle Eastern allies decide to handle the Syrian issue will be a major question during meetings in the coming days. Beside Mr. Hollande’s planned trips, Mr. Lavrov of Russia plans a trip next week to Turkey to discuss similar issues.

Qatar, Turkey and Saudi Arabia see Mr. Assad’s departure as the cornerstone of any effort to defeat Islamic State. They back an array of rebel groups in Syria, including Islamist groups collaborating with the al Qaeda-linked Nusra Front. They advocate the creation of no-fly zones in Syria immediately. For these Sunni states the presence of the Assad regime and the efforts of Iran and its Shiite militias to defend it with Russia’s help is a driver of the extremism among Sunnis that give sustenance to groups like Islamic State.

Other Gulf countries such as Kuwait and the United Arab Emirates are more concerned about the end of the conflict and less about the fate of Mr. Assad. Oman remains friendly to the Assad regime and officials from both countries meet regularly.

In the midst of Mr. Obama’s swing through Asia, White House officials said they have been focused on turning other countries’ newfound resolve to fight extremists into concrete commitments.

“That clearly will be a focal point,” said Ben Rhodes, a deputy national-security adviser to Mr. Obama. “What are the additional French contributions to this effort? How does that fit into the broader coalition effort? And then, what can they do together to work with other countries that are feeling a greater sense of urgency after Paris to do more?”

The United Nations Security Council pledged greater unity as it began deliberations on possible resolutions for countering Islamic State, but are considering two competing proposals to guide international action: one offered by Russia and a second circulated on Thursday by France.

The Security Council was expected to move quickly to review and vote on the French proposal. British Ambassador to the U.N. Matthew Rycroft, who holds the presidency of the Council this month, said the Russian proposal “seeks to legitimize the authority of Assad,” but he expressed the U.K.’s full support for the French proposal.

Casting doubt on the likelihood of a quick agreement, Mr. Obama said in Manila that it might take some months for Russia and Iran to come around on Mr. Assad. Russia’s military intervention in Syria has been aimed at shoring up Mr. Assad. But after previously directing its firepower predominantly at moderate rebels, Moscow this week deployed long-range bombers to hit Islamic State targets.

Mr. Rhodes said Russia has shown an interest in joining an effort to counter Islamic State—a prospect he deemed potentially constructive. But first, Russia must follow through, not just by continuing to strike Islamic State targets but also by ending strikes against groups that could be part of the moderate opposition in Syria, he said.

Mr. Rhodes laid out a road map for the Russians.

“What we would like to see is a focus militarily from Russia on [Islamic State], a focus through the political process and through their military restraint in preserving a moderate opposition that can be a part of a political process, and ultimately, Russia recognizing that the transition plan that has been set at Vienna will have to include at some point Assad stepping aside,” he said, referring to rounds of diplomatic talks being held in the Austrian capital. “Those are the elements that we want to see.”

Mr. Obama in meetings with foreign leaders during summits in Turkey and Asia during the last several days has aimed to elicit concrete, new commitments, but there was little in the way of immediate results.

After meeting with Mr. Obama on Thursday, Canadian Prime Minister Justin Trudeau said he would make good on a campaign promise to remove Canada’s jet fighters from the U.S.-led air campaign against Islamic State.

“We’re going to be discussing further the best ways Canada can move forward,” Mr. Trudeau said. “But I made a clear commitment to Canadians to withdraw the six jet fighters, and we will be doing that in collaboration and coordination with all of our allies to ensure that the coalition still has tremendous impact against” Islamic State.