F.B.I. and Justice Department officials have been quietly meeting with security researchers who have been working on approaches to provide such "extraordinary access" to encrypted devices, according to people familiar with the talks.

Based on that research, Justice Department officials are convinced that mechanisms allowing access to the data can be engineered without intolerably weakening the devices' security against hacking.

Against that backdrop, law enforcement officials have revived talks inside the executive branch over whether to ask Congress to enact legislation mandating the access mechanisms. The Trump White House circulated a memo last month among security and economic agencies outlining ways to think about solving the problem, officials said.

The F.B.I. has been agitating for versions of such a mandate since 2010, complaining that the spreading use of encryption is eroding investigators' ability to carry out wiretap orders and search warrants — a problem it calls "going dark."