“On the issue of U.S. person versus non-U.S. person, that’s an issue we’re giving a lot of thought to now,” Robert Litt, the general counsel to the director of national intelligence, told an American Bar Association conference last week. “That doesn’t mean that we have no protection for non-U.S. persons,” he said, noting that the main protection was that data had to be collected for “a valid foreign intelligence purpose.” But that is a standard the intelligence agencies can define for themselves in the case of foreigners.

Mr. Litt said that the government is now “giving some thought to whether there are ways that we can both introduce a little more rigor into that requirement.” But another American official said there were concerns about whether a decision to effectively extend the constitutional protections of the Fourth Amendment to some foreigners would create a precedent the government might later regret.

So far, the sharpest public criticism of the N.S.A. from within the administration has come from one of the chief clients for its intelligence reports: Secretary of State John Kerry. “The president and I have learned of some things that have been happening in many ways on an automatic pilot, because the technology is there and the ability is there,” Mr. Kerry said last week, adding that “some of these actions have reached too far.”

A senior administration official said that Mr. Kerry’s “automatic pilot” reference “went beyond our talking points,” but added that the president agreed and “has already made some decisions,” which have not been announced.

The administration’s reviews are being conducted in secrecy in part because of the secret nature of the N.S.A.’s operations. Initially, the reviews focused on domestic “bulk collection” programs begun after the Sept. 11, 2001, attacks, which eventually led to the N.S.A. program to collect the billing records of all calls, and, for a while, to collect a large volume of emails as well. (The email program ended, the N.S.A. says, in 2011.) In an interview last month, General Alexander said he was “open” to any alternative to having the government maintain that database of calls.

But General Alexander’s deputy, John C. Inglis, who has spent nearly three decades at the N.S.A. focused on the technology of intercepting and decoding foreign communications, told Congress last week that so far there was no satisfying alternative to a government library of calls and, seemingly by extension, text messages and many Internet searches.