Gen. Joseph F. Dunford Jr., the chairman of the Joint Chiefs of Staff, told lawmakers that he had sent a message through an intermediary to Qassim Suleimani, the commander of Iran’s paramilitary forces, that he would be held accountable for any attacks on the United States, according to officials who were in the briefings and described them on condition of anonymity.

In a radio interview, Mr. Pompeo said that the United States had not determined who was responsible for sabotage attacks last week on oil tankers in the Middle East, but that “it seems like it’s quite possible that Iran was behind” them.

He also defended the administration’s steps against Iran and said the United States would continue to “work to deter Iran from misbehavior in the region.”

“We’ve made clear that we will not allow Iran to hide behind its proxy forces, but that if American interests are attacked, whether by Iran directly or through its proxy forces, we will respond in an appropriate way against Iran,” Mr. Pompeo told Hugh Hewitt, a conservative radio host.

Like many things in Washington, reactions to the administration’s handling of the tensions with Iran have fallen along a sharp partisan divide.

Senator Lindsey Graham, Republican of South Carolina, told reporters that the briefing made clear that “this was an escalation of the threat stream we have not seen before.”

“I’m convinced the threat stream that they picked up is real and the actions of the Iranians went to a new level,” said Mr. Graham, who sits on the Foreign Relations Committee. “This is a defining moment for the Trump administration, the Trump administration needs to let them know that the maximum pressure campaign will continue.”