In June, after Mr. Trump said he would bar Muslims from entering the United States, President Recep Tayyip Erdogan of Turkey demanded that Mr. Trump’s name be removed from Trump Towers Istanbul. Mr. Erdogan, an Islamist, has arrested or fired 100,000 opponents and jailed 40,000 more after an unsuccessful military coup last summer.

After Mr. Trump’s election, Mr. Erdogan shifted his stance. “I believe we will reach a consensus with Mr. Trump, particularly on regional issues,” Mr. Erdogan said this month during a meeting with Turkish diplomats. Some cracks appeared to show after Mr. Trump’s immigration order Friday: Mr. Erdogan called the move “frankly disturbing.” But he said he would still meet with Mr. Trump at an unspecified date and raise the issue then.

And the name of Trump Towers remains unchanged.

In Egypt, President Abdel Fattah el-Sisi, who has smarted from American criticism about the country’s human rights record and about the military coup that brought him to power, appears to be welcoming Mr. Trump’s leadership. In December, after a phone call from Mr. Trump, Mr. Sisi agreed to delay a vote in the United Nations Security Council on Israeli settlements.

Mr. Sisi’s silence in the face of the executive order on refugees was conspicuous, despite widespread sentiment in the region that it was anti-Muslim.

In Kazakhstan, the country’s “president for life,” Nursultan A. Nazarbayev, whose poor human rights record is well documented, said Mr. Trump had called him in December and complimented him on the “miracle” he had wrought in his country over its 25 years of independence. Mr. Trump was apparently not referring to Mr. Nazarbayev’s 2015 re-election, which the Kazakh leader won with 97.7 percent of the vote.