CBSSports.com senior writer Danny Knobler reported earlier the Rays were close to signing Damon, 37. Damon hit .271/.355/.401 for the Tigers in 2010, but managed just eight home runs.
Ramirez hit .298/.408/.460 with the Dodgers and White Sox last season with nine homers, appearing in just 90 games.
With Ramirez in the fold, Damon will likely have to man the outfield some for the Rays and can also serve as a babysitter for his former Red Sox teammate.
UPDATE: Damon will get $5.25 million from the Rays, plus $750,000 in attendance bonuses. Ramirez will earn $2 million, Heyman tweets .
-- C. Trent Rosecrans