If Bell declines, the Padres will collect two supplemental picks in June's First-Year Player Draft as compensation for Bell signing elsewhere.
Byrnes has said the team will likely offer the three-time All-Star arbitration.
The Padres' payroll for next season projects at $54 million, CEO and vice chairman Jeff Moorad said last month. If Bell returns to San Diego on a one-year deal, the team won't have nearly as much money to spend this offseason.
On Tuesday, Bell didn't sound hopeful that a return to San Diego would be in the cards.
"I think it's a long shot," Bell said.
But that doesn't mean Bell will make a decision before the Dec. 7 deadline. In fact, he plans on waiting until the very end to make his decision.
"I'm keeping my options open. I'm going to see what happens. I'm going to wait until the very end to see," Bell said.
"Who knows? Maybe at the final hour they'll [Padres] come and offer me something [a multi-year deal]. I know that Jed and I had some good talks, and now he's in Chicago. I haven't talked to Josh yet."
Jed Hoyer is the former Padres general manager who now holds that same post with the Cubs.
Bell, who has more saves since the start of the 2009 season (132) than any other Major League pitcher, would like to remain with the Padres but realizes that his future -- and a brighter financial future, too -- might be elsewhere.
"I would prefer to stay on the West Coast because my family is here, but I'm fine playing on the East Coast," Bell said. "I've talked to my agent, and there could be opportunities with some of those teams."
Bell's agents, Seth and Sam Levinson, recently negotiated a big deal for closer Jonathan Papelbon. Papelbon agreed to a four-year contract with the Phillies for $50 million that also calls for a $15 million fifth-year vesting option.