WASHINGTON — In an investigation involving guns and drugs, the Justice Department obtained a court order this summer demanding that Apple turn over, in real time, text messages between suspects using iPhones.

Apple’s response: Its iMessage system was encrypted and the company could not comply.

Government officials had warned for months that this type of standoff was inevitable as technology companies like Apple and Google embraced tougher encryption. The case, coming after several others in which similar requests were rebuffed, prompted some senior Justice Department and F.B.I. officials to advocate taking Apple to court, several current and former law enforcement officials said.

While that prospect has been shelved for now, the Justice Department is engaged in a court dispute with another tech company, Microsoft. The case, which goes before a federal appeals court in New York on Wednesday and is being closely watched by industry officials and civil liberties advocates, began when the company refused to comply with a warrant in December 2013 for emails from a drug trafficking suspect. Microsoft said federal officials would have to get an order from an Irish court, because the emails were stored on servers in Dublin.

The conflicts with Apple and Microsoft reflect heightened corporate resistance, in the post-Edward J. Snowden era, by American technology companies intent on demonstrating that they are trying to protect customer information.

“It’s become all wrapped up in Snowden and privacy issues,” said George J. Terwilliger III, a lawyer who represents technology companies and as a Justice Department official two decades ago faced the challenge of how to wiretap phone networks that were becoming more digital.

President Obama has charged White House Homeland Security and cybersecurity officials, along with those at the Justice Department, the F.B.I. and the intelligence agencies, with proposing solutions — some legislative, some not — to the technology access issue. They are still hashing out their differences, according to law enforcement and administration officials.

Some Justice and F.B.I. officials have been frustrated that the White House has not moved more quickly or been more outspoken in the public relations fight that the tech companies appear to be winning, the law enforcement officials said, speaking only on condition of anonymity because they were not authorized to discuss the private conversations.

The White House, after months of study, has yet to articulate a public response to the argument that a victory in the Microsoft case would provide authoritarian governments, particularly the Chinese and Russians, with a way to get access into computer servers located in the United States.

“Clearly, if the U.S. government wins, the door is open for other governments to reach into data centers in the U.S.,” Brad Smith, Microsoft’s general counsel, said in a recent interview. Companies and civil liberties groups have been sending in briefs of their own, largely opposing the government’s surveillance powers.

At issue are two types of encoding. The first is end-to-end encryption, which Apple uses in its iMessage system and FaceTime, the video conversation system. Companies like Open Whisper Systems, the maker of Signal, and WhatsApp have adopted such encryption for stand-alone apps, which are of particular concern to counterterrorism investigators.

With Apple, the encryption and decryption is done by the phones at either end of the conversation; Apple does not keep copies of the message unless one of the users loads it into iCloud, where it is not encrypted. (In the drug and gun investigation this summer, Apple eventually turned over some stored iCloud messages. While they were not the real-time texts the government most wanted, officials said they saw it as a sign of cooperation.)

The second type of encoding involves sophisticated encryption software on Apple and Android phones, which makes it all but impossible for anyone except the user of the phone to open stored content — pictures, contacts, saved text messages and more — without an access code. The F.B.I. and local authorities oppose the technology, saying it put them at risk of “going dark” on communications between terrorists and about criminal activity on city streets. The American military is more divided on the issue, depending on the mission.

Officials say a court fight with Apple is still an option, though they acknowledge it would be a long shot. Some object that a legal battle would make it harder for the companies to compromise, the law enforcement officials said. They added that Apple and other companies have privately expressed willingness to find common ground.

Apple declined to comment on the case for this article. But company officials have argued publicly that the access the government wants could be exploited by hackers and endanger privacy.

“There’s another attack on our civil liberties that we see heating up every day — it’s the battle over encryption,” Tim Cook, the company’s chief executive, told a conference on electronic privacy this year. “We think this is incredibly dangerous.”

Echoing the arguments of industry experts, he added, “If you put a key under the mat for the cops, a burglar can find it, too.” If criminals or countries “know there’s a key hidden somewhere, they won’t stop until they find it,” he concluded.

The Microsoft case centers on whether the fact that data is stored around the world relieves American firms of turning it over. The government, which won in Federal District Court, has argued in its brief to the appeals court that where the data is stored is irrelevant because the company still has control of email records. The White House declined to comment because the case is in litigation.

“People want to know what law will be applied to their data,” Mr. Smith of Microsoft said. “French want their rights under French law, and Brazilians under Brazilian law. What is the U.S. government going to do when other governments reach into the U.S. data centers, without notifying the U.S. government?”

Chinese firms already have plans to build facilities on American soil that would store electronic communications, so the question may be more than hypothetical. In its brief, Microsoft argues that Congress will ultimately have to weigh in on the issue, since it is as much a political matter as a legal one: “Only Congress has the institutional competence and constitutional authority to balance law enforcement needs against our nation’s sovereignty, the privacy of its citizens and the competitiveness of its industry.”