The offer, which hasn’t previously been reported, was declined by the Obama administration. But it shows the degree to which Mr. Snowden’s revelations have shaken the NSA’s foundations—unlike any event in its six-decade history, including the blowback against domestic spying in the 1970s.

The post-Snowden era has forced a major re-evaluation of NSA operations by the administration and on Capitol Hill, and the review is likely to alter the agency’s rules of the road. “It was cataclysmic,” Richard Ledgett, who heads a special NSA Snowden response team, said of the disclosures. “This is the hardest problem we’ve had to face in 62 years of existence.”

Broad new controls, though, run the risk of overcorrecting, leaving the agency unable to respond to a future crisis, critics of the expected changes warn.

When the leaks began, some top administration officials found their confidence in Gen. Alexander shaken because he presided over a grave security lapse, a former senior defense official said. But the officials also didn’t think his resignation would solve the security problem and were concerned that letting him leave would wrongly hand Mr. Snowden a win, the former defense official said.