The judge in the case, being handled in federal court in Riverside, California, scheduled a hearing for late afternoon on Monday to consider the request.
The unexpected development in the high-profile case raised questions as to whether the Justice Department might be backing off from its confrontation with Apple.
In a court filing, the Justice Department said the new technique came to light on Sunday, but it provided no further details.
The government has obtained a court order requiring Apple to write new software to disable passcode protections on a phone used by one of the shooters in the December attack in San Bernardino, California.
Apple, with the backing of much of the tech industry, is fighting the order, contending that it will undermine computer security and privacy for all consumers.
(Reporting by Mari Saito and Joseph Menn; Editing by Bill Rigby and Jonathan Weber)