WASHINGTON—Shortly after former government contractor Edward Snowden revealed himself in June as the source of leaked National Security Agency documents, the agency's director, Gen. Keith Alexander, offered to resign, according to a senior U.S. official.

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...