According to BuzzFeed News, Apple claims to have offered the government four alternatives that would not involve the need for a backdoor, one of which involved triggering an iCloud backup when connected to a known WiFi network. However it was later discovered that they were unable to do this as someone had apparently changed the Apple ID password associated with the device.
Apparently what happened is that less than 24 hours after the government took possession of the device, someone had reset the password on the device, meaning that had someone not done that, the FBI could have easily accessed the phone themselves. The FBI is claiming that someone at the San Bernardino Health Department was behind the password change, presumably in an attempt to try and crack the phone themselves.
To that end, Apple claims that because of this alleged screw-up, granting the FBI backdoor access to the iPhone would be unnecessary following the password change. That being said, it is unlikely that the FBI will accept this explanation, so we can only expect this saga to keep moving forward until something gives.