If a president does not brief his staff, intelligence agencies have few options to learn about the meeting. Their most obvious solution — eavesdropping — is off limits when it comes to the commander in chief, even during a meeting with the leader of an adversary, according to former intelligence officers.

Still, the intelligence agencies would probably try to intercept Russian discussions of what was said in the meeting between Mr. Trump and Mr. Putin, former officials said.

The disconnect between the White House and intelligence agencies could create a thorny situation if American spies collect information that might be embarrassing to Mr. Trump — such as Russian officials saying that Mr. Putin had extracted concessions from Mr. Trump during the Helsinki meeting.