The result is that on his stops in Egypt, Kuwait, Jordan, Lebanon and Turkey, Mr. Tillerson will either be delivering difficult messages — urging partners to follow through on promises — or trying to assuage furious allies. And while previous administrations had papered over differences with generous promises of financial aid, Mr. Tillerson’s goody bag will be all but empty.

Still, Mr. Tillerson will face those considerable challenges newly fortified. His standing appears to have risen in the White House after a year in which his tenuous relationship with the president undermined his diplomatic efforts, as senior administration officials predicted his imminent departure and foreign leaders quietly wondered whether he actually spoke for his mercurial boss.

But he still must cope with the president’s intemperate remarks about other countries, and he is forgoing a visit to Jerusalem, perhaps because Jared Kushner, the president’s son-in-law and senior adviser, is still Israel’s principal American intermediary.

Mr. Tillerson begins his official visits on Monday in Cairo, but the most difficult stop on his itinerary will be Ankara. The Turkish president, Recep Tayyip Erdogan, has threatened to attack the Syrian city of Manbij, where American forces and their Kurdish allies are entrenched after evicting the Islamic State. Turkey views the Kurdish forces as part of a violent insurgency within Turkey.

“They tell us, ‘Don’t come to Manbij.’ We will come to Manbij to hand over these territories to their rightful owners,” Mr. Erdogan said Tuesday in a speech to his party. As for American assurances that arms given to the Kurds will be used only to fight the Islamic State, “don’t expect us to believe,” Mr. Erdogan said. “We don’t buy it.”

Turkey, a NATO ally, has allowed American forces to use Incirlik Air Base as a crucial staging ground for the air campaign over Syria. But Mr. Erdogan’s creeping authoritarianism, his security detail’s attack in Washington on peaceful protesters, his saber-rattling on Syria and Turkey’s arrests of American citizens and State Department employees have left the relationship severely strained.

For the Turks, fury at the administration’s refusal to begin the process of extraditing the Muslim cleric Fethullah Gulen, who Turkey says orchestrated a 2016 coup from his self-imposed exile in Pennsylvania, compounds the anger at American support for Kurdish forces. On Thursday, Serkan Golge, an American citizen and NASA scientist, was sentenced to seven and a half years in prison for alleged links to Mr. Gullen, although no proof of such links was offered in court. The State Department said it was “deeply concerned” by the decision.