The deal is believed to be worth $3 million in base salary with a possible $2 million in incentives.
Andrew Friedman, Rays executive vice president of baseball operations, confirmed the two sides are negotiating.
"We are still working through some of the finer points," said Friedman. "We are hopeful there will be a positive announcement early next week."
Since trading Delmon Young to the Twins, the Rays have been looking for a left-handed bat who can play right field and DH. Floyd, 35, has a .279 career batting average with 222 home runs in 15 Major League seasons with the Expos, Marlins, Red Sox, Mets and Cubs. He hit .284 with nine home runs for the Cubs in 2007.
Because Floyd has a history of injuries, the report says an agreement with the Rays won't happen until he passes a physical.
If Floyd signs with the Rays, he would be in the mix to compete for the right field and DH jobs with Jonny Gomes and Rocco Baldelli, assuming the latter is healthy in 2008.
Bill Chastain is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.