As the Obama administration insists that Congress was fully informed about the National Security Agency's widespread surveillance on Americans' phone records, its intelligence chief is becoming a complication.

James Clapper, the director of national intelligence, has now admitted he gave the "least untruthful" answer to a direct question in March about the extent of surveillance on US citizens. The admission sets up a critical test of Clapper's relationship with the congressional committees that oversee him – committees the Obama administration is relying on for its defense of the surveillance efforts.

The Obama team is expressing support for Clapper as criticism of him mounts. "The president has full faith in director Clapper and his leadership of the intelligence community," National Security Council spokeswoman Caitlin Hayden told the Guardian on Wednesday.

At least one member of Congress is calling for Clapper's head. On his Facebook page, Justin Amash, a Michigan Republican, wrote that Clapper "lied under oath" to Congress.

"It now appears clear that the director of national intelligence, James Clapper, lied under oath to Congress and the American people," Amash posted on Wednesday morning. "Members of Congress can't make informed decisions on intelligence issues when the head of the intelligence community wilfully makes false statements. Perjury is a serious crime. Mr Clapper should resign immediately."

At a hearing of the Senate intelligence committee on 12 March, Oregon Democrat Ron Wyden grew frustrated that he could not get a "direct answer" from Clapper about a question Wyden said he had been posing to the intelligence agencies in a series of letters for a year: when do US spies need a warrant to surveil Americans' communications?

"What I wanted to see is if you could give me a yes or no answer to the question: does the NSA collect any type of data at all on millions or hundreds of millions of Americans?" Wyden asked Clapper.

"No, sir," Clapper said. "Not wittingly. There are cases where they could inadvertently, perhaps, collect, but not wittingly." (...)