The tight end has been skipping workouts with the team in recent weeks in a bid to force contract talks with the NFC West side.
The 30-year-old told the Atlanta Journal-Constitution: "It's not the end of the world if I don't get [a new contract]. If I get it, that's good. It's moving slowly, but surely.
"At the end of the day, if I don't get the deal, at least I tried and I put my foot down. If I don't get it at the moment, I'm not going to lose any sleep."
Davis, who is set to earn $4.7m (£2.75m) in 2014 and has two years remaining on his contract, had 850 receiving yards and 13 touchdowns for the 49ers last season.