The English version of the German news weekly Der Spiegel quoted the Arizona Republican as saying, "Of course, they should resign or be fired" after he was asked if Alexander, a four-star Army general, should resign because of alleged spy-agency shortcomings that led to U.S. surveillance leaks by former NSA contractor Edward Snowden.
Some news organizations quoted McCain as responding to the question by saying "he should resign," suggesting Alexander, rather than "they should resign."
"Senator McCain believes that there needs to be accountability for the Snowden leaks, but he is not calling for the resignation of General Alexander, who is retiring soon," McCain's office said late Sunday.
Alexander is due to retire in March. He has led the NSA since 2005.
In the interview, McCain spoke about spy-agency changes since the Sept. 11, 2001, terrorist attacks that let low-level people get high-level security clearances.
"Why did Edward Snowden have that information?" McCain asked, pointing out Snowden wasn't even a government employee. "And what are we doing as far as screening people who have access to this information? It's outrageous, and someone ought to be held accountable."
He was asked who should be held accountable.
"The head of the NSA, the president of the United States, the congressional intelligence committees, all of these contractors we pay that were responsible for performing the background checks. There should be a wholesale housecleaning," McCain said.
He was then asked if Alexander should resign.
"Of course, they should resign or be fired. We no longer hold anybody accountable in Washington," Der Spiegel quoted McCain as saying.
McCain also said he believed Snowden would "never" return to the United States, where he faces charges of espionage and theft of government property.
Snowden is living in Russia after being granted a temporary but extendable one-year asylum Aug. 1.
"[Russian] President Vladimir Putin will grant him asylum indefinitely," McCain said. "The Russians know if they send him back that that's a lesson to other people who might defect. I'm sure that Mr. Snowden has told them everything that he possibly knows."
When Der Spiegel said Snowden denied passing any information to Moscow, McCain said, "If you believe that Mr. Snowden didn't give the Russians information that he has, then you believe that pigs can fly."