“In terms of whether I’d ever pick the phone up, I’ve never shut off the phone to anyone that wants to talk or have a productive conversation,” Mr. Tillerson said.

The Trump administration appears to have grasped an important point about Iran: The very thing that the administration complains and worries about — Iran’s expanding influence in the region — makes it imperative that the two countries maintain at least a working relationship.

The United States will have a hard time solving problems in the Mideast without Tehran’s cooperation: in Lebanon, where it backs the Shiite militant group Hezbollah; in Syria, where it is propping up the government of President Bashar al-Assad; in Iraq, where it supports the government of Prime Minister Haider al-Abadi and trains powerful Shiite militias; and in Yemen, where to some extent, it is backing the Houthi rebels against the government.

That is why some analysts are saying there would be room for compromise if the two countries ever sat down to talk, with the United States trading its control over the Iranian economy for concessions in regional affairs from Iran.

The Trump administration did impose a series of sanctions on individuals and companies after Iran conducted a missile test in February. But that was in line with policies set out by President Obama.

And on Thursday, the Senate Foreign Relations Committee approved a sweeping set of sanctions against Iran for human rights violations and its support of terrorism.

The senators did so over the objections of the former secretary of state, John Kerry, who negotiated the nuclear deal. In a series of Twitter posts on Wednesday, Mr. Kerry called on the Senate committee to refrain from imposing sanctions and warned against “confrontation without conversation.”