The gun is not by chance over 50 years old and a C&R? The FFL did not give any other information, like the buyer he has lined up happens to be a LEO? It doesn't happen to be an exempt olympic-style target shooting pistol, or a cowboy-style single action revolver?
Are you sure the person is really an FFL?
Featureless' answer is completely correct: The FFL could buy the gun from you, but he can not sell it to most people (except for LEOs, olympic pistols, C&R, and single actions). Used guns are explicitly NOT exempt from this. The FFL would just have to sit on it. But every gun dealer in California knows this, it is drummed into them at all times. This makes me wonder whether this party is a real FFL, or perhaps a completely incompetent one (who is probably about to go out of business), or a fraud, or a CA DoJ or federal ATF sting operation, or something else got screwed up here. If this is a normal 01-FFL (gun store), then this situation is just not realistic.
Perhaps you talked to the youngest apprentice at a gun store, and the guy is about to be fired?