WASHINGTON—President-elect Donald Trump has named Exxon Mobil Corp. Chief Executive Rex Tillerson as his secretary of state, a transition official said Monday, picking a veteran CEO who has had extensive overseas business dealings but whose relationships with foreign leaders could complicate his confirmation prospects.

Mr. Tillerson was a comparatively late entry in the secretary of state competition, but he impressed the president-elect as a successful deal-maker in what one transition aide called the “Trumpian” mold.

If confirmed by the U.S. Senate, Mr. Tillerson will be the public face of a diplomatic approach that envisions more cooperation with Russia and concessions from China on trade and security matters.

Trump’s Inner Circle

“His tenacity, broad experience and deep understanding of geopolitics make him an excellent choice for Secretary of State,” Mr. Trump said in a statement. “He will promote regional stability and focus on the core national security interests of the United States. Rex knows how to manage a global enterprise, which is crucial to running a successful State Department, and his relationships with leaders all over the world are second to none.

Mr. Trump injected a bit of theater into what is normally a staid and behind-the-scenes process, offering personal impressions of the candidates and tweeting out his timetable for a decision.

On Sunday, he tweeted that Mr. Tillerson is a “world class player and dealmaker.”

“Stay tuned!” he wrote.

Word of Mr. Trump’s selection began leaking out Monday night.

One of the finalists, 2012 Republican presidential nominee Mitt Romney, wrote a message on Facebook that suggested he was out of the running.

Mr. Romney wrote: “It was an honor to have been considered for secretary of state of our great country.”

In choosing Mr. Tillerson, the president-elect passed over various campaign allies and established political figures. Among those he considered—and rejected—were former New York City Mayor Rudy Giuliani, one of his closest campaign advisers; U.S. Sen. Bob Corker (R., Tenn.), chairman of the Senate Foreign Relations Committee; and David Petraeus, a former director of the CIA.

In the days after the election, Mr. Giuliani was seen as the top contender. But Mr. Giuliani, one of Mr. Trump’s most loyal campaign supporters, took himself out of contention on Nov. 29, Mr. Trump said.

Mr. Corker was the most traditional candidate. The current and previous secretaries of state, John Kerry and Hillary Clinton, came out of the Senate.

Gen. Petraeus, a former commander of troops in Iraq and Afghanistan, would have brought extensive military and intelligence experience to the job. He was director of the Central Intelligence Agency during the Obama administration. But Gen. Petraeus also faced scrutiny of his 2015 guilty plea to a misdemeanor charge of mishandling classified material in a case involving his biographer, with whom he said he had an extramarital affair.

In addition, John Bolton, who served as U.S. ambassador to the United Nations in the George W. Bush administration, was a candidate with a background in diplomatic service. Also during the Bush administration, Mr. Bolton served as the State Department’s counterproliferation czar and focused specifically on the Iranian and North Korean threats. Yet he’s seen as one of the most hawkish members of the Republican Party in recent decades, advocating for the U.S. to bomb Iranian nuclear sites and opposing any diplomatic negotiations with North Korea. He could have been at odds with Mr. Trump on some issues, given the president-elect has suggested the U.S. should pull back from overseas intervention.

Mr. Tillerson’s nomination faces bipartisan resistance in the Senate over his ties to Russian President Vladimir Putin.

His company has long done business in Russia. He has known Mr. Putin since he represented Exxon’s interests in Russia during the regime of Boris Yeltsin. In a sign of the close relationship, the Kremlin bestowed the country’s Order of Friendship decoration on Mr. Tillerson after he struck a 2011 deal that gave Exxon access to prized Arctic resources and allowed Russian state oil company OAO Rosneft to invest in Exxon concessions around the world.

Mr. Tillerson’s past opposition to sanctions on Russia is likely to trigger blowback among Senate Republicans, many of whom have rejected Mr. Trump’s more conciliatory stance toward the country and its president. No Senate Republicans have yet said they would vote against Mr. Tillerson. But a number of senators expressed reservations. Sen. Marco Rubio (R., Fla.), a member of the Foreign Relations Committee, the panel that would hold confirmation hearings on the nomination, said in a tweet Sunday that “being a ‘friend of Vladimir’ is not an attribute I am hoping for” in the next secretary of state.

Mr. Trump’s secretary of state will have to navigate a host of high-stakes foreign policy challenges across the globe. The country’s top diplomat also will be tasked with carrying out Mr. Trump’s vision for the U.S. role on the world stage, which is so far not entirely clear.

A Trump administration will quickly have to contend with a volatile Middle East. Military involvement by Russia and Iran to boost the Assad regime in Syria has complicated the fight against Islamic State. The next secretary of state would be at the forefront of any negotiations with Russia on a resolution to the Syrian conflict while also tending to U.S. allies in the region who oppose the Assad regime.

Mr. Trump’s approach to U.S. relations with Russia will be one of his most closely watched moves, given his comments about Russian President Vladimir Putin and U.S. intelligence assessment that Moscow used cyberattacks to try to help Mr. Trump in the election. Republican leaders in Congress recently have expressed deep misgivings about any warming to Mr. Putin.

Exxon Mobil CEO Rex Tillerson has been picked by Donald Trump to serve as secretary of state. Photo: LM Otero/Associated Press

Mr. Trump also has signaled that he wants to implement a more adversarial relationship with China, challenging Beijing on trade and security measures. Mr. Trump’s relations with China are already off to a rocky start, with Beijing balking at his protocol-breaking phone call with the president of Taiwan.

Mr. Trump will need China, however, in efforts to address the growing threat from North Korea. China is seen as one of the few countries in the world with influence over Pyongyang. President Barack Obama told Mr. Trump that he believes North Korea is the biggest foreign policy challenge he faces once in office.

He’ll have to decide whether to adhere to the international Paris climate change agreement and to continue with the re-establishment of U.S. relations with Cuba. Both were among Mr. Obama’s top foreign policy initiatives and have drawn opposition from Republicans.

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Mr. Trump hasn't specifically said whether he will roll back the deal the U.S. and other world powers reached with Iran to restrain its nuclear program, though he has been a fierce critic of the agreement. Israeli Prime Minister Benjamin Netanyahu has already begun to put pressure on Mr. Trump to back out of the deal, while Mr. Obama has tried to convince his successor that doing so would be a bad idea. U.S. relations with Israel have been deeply strained by disagreements between Messrs. Obama and Netanyahu on the Iran deal and other issues.

“I am honored by President-elect Trump’s nomination and share his vision for restoring the credibility of the United States’ foreign relations and advancing our country’s national security,” Mr. Tillerson said in a statement. “We must focus on strengthening our alliances, pursuing shared national interests and enhancing the strength, security and sovereignty of the United States.”

—Bradley Olson and Kristina Peterson contributed to this article

Write to Peter Nicholas at peter.nicholas@wsj.com and Carol E. Lee at carol.lee@wsj.com