At a political rally on Sunday, he displayed a piece of paper that he took from his pocket, which he said was evidence of a foreign conspiracy but did not say by which country. He has revealed few other details about the letter and has not provided a copy to Parliament or to the news media.
In his speech on Thursday, he identified the threatening country as the United States, which has long been a favorite target of his political speeches.
He said that American officials said that Pakistan “will have to suffer” if he remained as prime minister.
“No reason was stated,” he said. “They are treating Pakistan as if we are their slaves.”
American officials dismissed Mr. Khan’s characterization of events.
“There is no truth to these allegations,” a State Department spokeswoman said Thursday, speaking on condition of anonymity in line with diplomatic protocol.
Opposition politicians have cast doubt on the letter’s authenticity and Pakistani security officials have dismissed it as an internal diplomatic communication that Mr. Khan has misrepresented.
The political crisis comes as Pakistan, home to 220 million people and the second-largest Muslim population in the world, wrestles with double-digit inflation that has pushed the cost of basic goods, like food and fuel, beyond many Pakistanis’ reach. The economic challenges have fueled criticism that Mr. Khan has mismanaged the economy and failed to deliver on his touchstone promise of creating an Islamist welfare state.