The 28-year-old Gore has one year and $5 million remaining on his contract. Gore's agent, Drew Rosenhaus, could not be reached for comment.
Gore believes he must try to get a new deal now, after he fractured his hip last season. He already has undergone two knee surgeries and two shoulder surgeries.
San Francisco was planning to lean on Gore, with Alex Smith the likely starting quarterback. With that leverage, Gore believes the time to get his last lucrative NFL deal is now.
Adam Schefter is ESPN's NFL Insider.