WASHINGTON—A confluence of crises in Syria and North Korea is forcing President Donald Trump to re-evaluate his fledgling foreign policy, deciding which advisers he will listen to and which campaign pledges to jettison.

The apparent chemical-weapons attack in Syria and the latest ballistic missile test by North Korea raise the stakes for two upcoming events: Mr. Trump’s summit this week with Chinese President Xi Jinping , a key ally of North Korea, and Secretary of State Rex Tillerson’s planned visit next week to Russia, a patron of the Syrian regime.

The president has signaled in recent days that he is willing to scramble the U.S. approach to North Korea, including by using military force, and he put his stance on Syria in play as well in a news conference on Wednesday.

Mr. Trump thus showed a readiness to shift from positions he held during his presidential campaign, when he entertained the prospect of talks with North Koreans and others. But the crises also will help indicate whether there has been a change in how—and by whom—the administration’s foreign policy is developed.

On Wednesday Mr. Trump approved changes that removed adviser Steve Bannon from the operations of the White House National Security Council and restored the roles of traditional U.S. security officials. The administration portrayed the move as long-planned, but others said it was the result of shifting White House dynamics.

“It looks like it’s snapping back to the mean of more regular order in the foreign policy-making process,” said Richard Fontaine, president of the Center for a New American Security and a former foreign-policy adviser to Sen. John McCain (R., Ariz.). “The question is going to be how does all of this connect with the president and rest of the White House?”

In recent months, Mr. Trump roiled U.S. ties with many allies through Twitter messages and public comments, eclipsing the more sober and cautious statements by senior advisers such as Defense Secretary Jim Mattis, Mr. Tillerson and others.

But the situations in North Korea and Syria require effective, reliable cooperation from key allies, such as Japan and South Korea, or European and Arab nations, said R. Nicholas Burns, a former career diplomat and undersecretary of state during the administration of President George W. Bush.

“This is a brutal experience; hopefully it will empower the pragmatic, expert part of the administration,” Mr. Burns said.

While Mr. Trump hasn’t yet laid out specific policy shifts, the developments this week in Syria and North Korea, and the visit of Mr. Xi to his Florida estate, have narrowed the window of time in which Mr. Trump has to act.

“People are going to be looking to see how the administration deals with these specific problems,” Mr. Fontaine said.

“The response to North Korea, to Xi, to Syria—they’re going to be read as having broader significance into the kind of foreign policy that the administration wants to conduct going forward. It heightens the importance of these three events,” he said.

Mr. Trump has yet to develop a broad policy on Syria, which overlaps with his strategy against Islamic State, his goals for relations with Russia and his efforts to engage the Persian Gulf states, which have long wanted the U.S. to force out Syrian President Bashar al-Assad.

Mr. Trump has filled the top slots in his foreign-policy and national-security team, but many other positions remain empty. Mr. Trump’s choice for ambassador to Israel, David Friedman, was recently sworn in, but his pick for ambassador to China hasn’t been confirmed; he has yet to nominate his choice for ambassador to Russia.

The administration’s most visible point person on Syria, and Russia, has been Nikki Haley, the U.S. ambassador to the United Nations. Ms. Haley said Wednesday the U.S. would act on its own if Russia doesn’t cooperate in a response to the chemical weapons attack.

On China, Mr. Trump’s son-in-law and senior adviser in the White House, Jared Kushner, has played a leading role in formulating the administration’s policy. The president’s top diplomat, Mr. Tillerson, also tried to lay groundwork for this week’s visit by Mr. Xi during a recent trip to Beijing.

Susan Thornton, the acting assistant secretary of state for East Asian and Pacific affairs, said Wednesday the Trump administration wants China to close off its financial pipeline to North Korea, because “they are going to have an important part to play in the international effort to do something about North Korea’s increasingly provocative moves.”

North Korea is “a big problem,” Mr. Trump said Wednesday. “We have somebody that is not doing the right thing, and that’s going to be my responsibility.” Yet, Mr. Trump said in an interview with the Financial Times this week that “if China is not going to solve North Korea, we will.”

Mr. Trump on Wednesday spoke by phone with Japan’s Prime Minister Shinzo Abe and said the U.S. would “continue to strengthen its ability to deter and defend itself and its allies with the full range of its military capabilities,” according to the White House.

Mr. Trump’s rhetoric toward China on North Korea in advance of his meeting with Mr. Xi could better position him, said Jeffrey Bader, who was President Barack Obama’s top adviser on Asia.

“The Chinese understand that we are looking at new more strenuous, more draconian actions,” Mr. Bader said. “In terms of preparation for the meeting with Xi, it is helpful for the Chinese to perceive that and should help motivate them to supportive of a tougher approach.”

Mr. Trump has otherwise taken a rather traditional U.S. approach to China—recognizing, for instance, the “One China” policy he initially said was up for negotiation—despite fiery campaign rhetoric against Beijing over its trade and economic policies. Mr. Trump has a host of other issues to discuss with Mr. Xi, including trade and maritime-security concerns in the South China Sea.

As a candidate, Mr. Trump promised to take strong action on trade and label Beijing a currency manipulator on his first day in office, but he has since backed off those positions.

Mr. Trump also took a more conciliatory tone toward Mr. Assad and North Korea’s leader Kim Jong Un. He said at one point Mr. Kim deserved credit for taking control of the country and that he would have no problem speaking with him. Mr. Trump said he didn’t like Mr. Assad but that he was fighting Islamic State.

While Mr. Trump criticized Mr. Obama throughout the campaign for drawing a red line on Syria and not enforcing it, he was against military action in Syria at that time. And he drew his own red line on Syria on Wednesday.

Mr. Trump’s swiftest policy turnaround has been on the future of Mr. Assad. Just days ago, his administration dropped the longtime U.S. demand that Mr. Assad must relinquish power as part of any political resolution to the Syrian conflict. On Wednesday Mr. Trump said the chemical attack had changed his view.

“It’s extraordinary to me that they could pivot so quickly,” said Aaron David Miller, an expert at the Wilson Center and former adviser to Republican and Democratic secretaries of State. “We’ve gone from risk aversion to risk readiness in 24 hours.”