In an implicit challenge to American policy against other adversaries, two top United Nations officials, Secretary General António Guterres and Michelle Bachelet, the high commissioner for human rights, called this week for the easing of financial sanctions against economically strapped countries so that they could confront the spreading virus. Ms. Bachelet said the economic penalties could impede medical treatment in Iran, Venezuela, Cuba, North Korea and Zimbabwe — all of which are penalized by the United States.

Speaking to reporters on March 22, Mr. Trump cast himself as providing relief to at least two of those nations. “On North Korea, Iran and others, we are open for helping other countries. It is a very serious time,” Mr. Trump said, adding that he had offered “a glad hand" to “many other countries.”

The United States thus far has committed $274 million to 64 countries and the United Nations refugee agency. Bonnie Glick, deputy administrator at the United States Agency for International Development, challenged other nations on Thursday to give “no-strings-attached contributions” to help the world combat the pandemic.

“In the battle against the coronavirus, we are saving lives, not saving face,” Ms. Glick said.

With Iran’s health care system swamped by one of the world’s worst outbreaks of the coronavirus, the State Department said last month that it was “prepared to assist the Iranian people in their response efforts,” a message conveyed through the Swiss government, which acts as an intermediary between Washington and Tehran.

The State Department could not clarify precisely what sort of assistance the Trump administration offered. But the gesture was a shift for an administration that has worked to undermine Iran’s government in every way it can, and which imposed new economic sanctions on Tehran as recently as last week.

Despite a swiftly mounting death toll, which has surpassed 2,000, Iran quickly rejected the American offer, making clear that what it really wants is broader relief from the sanctions Mr. Trump has imposed since he withdrew two years ago from the 2015 Iran nuclear deal. Even before the virus struck, Iran appealed to international opinion by arguing that the American sanctions were causing innocent people to suffer.

In an open letter to the American people last week, Iran’s president, Hassan Rouhani, said the United States was “aiding the spread of this virus with its sanctions,” which he said “have drastically undermined the ability of the Iranian people to fight the coronavirus and some among them are losing their lives as a result.”