"We've been talking," Horwits said. "It's starting to get serious. There have been offers, though, from multiple teams for Arthur."
One of those clubs competing with the Reds for Rhodes' services is the Cardinals, who made an offer last week. Horwits would not reveal the details about any of the offers made for Rhodes.
On Nov. 17, Affeldt signed a two-year, $8 million contract with the Giants. Reds general manager Walt Jocketty has since increased his efforts to find another veteran lefty and expressed interest in Rhodes, along with other free agents like Joe Beimel, Darren Oliver, Will Ohman and Dennys Reyes.
The Reds have one experienced lefty in the bullpen in Bill Bray. But they could be looking for more veteran help in general since free agent right-handers David Weathers and Mike Lincoln have yet to be re-signed. Weathers, a Type B free agent, was offered arbitration on Monday night.
Last season for the Mariners and Marlins, the 39-year-old Rhodes was 4-1 with a 2.04 ERA in 61 games. After a July 31 trade to Florida, he posted a 0.68 ERA in 25 games.
Overall, lefties batted .157 (11-for-70) against Rhodes, while right-handers batted .309 (17-for-55). Over his 17-year big league career, lefties have batted .221 against him, while righties have hit .245.
A Type B free agent, Rhodes was not offered arbitration by the Marlins, and there are no compensation concerns for any club that wants to sign him.
Mark Sheldon is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.