According to NESN, the Pirates received reliever Craig Hansen and outfielder Brandon Moss from the Red Sox. Pittsburgh also acquired third baseman Andy LaRoche and minor-league pitcher Bryan Morris from the Dodgers.
The reported deal came one day after Ramirez criticized the Red Sox in an interview with ESPNDeportes, claiming that he has been mistreated by the defending World Series champs.
Ramirez, 36, is batting .299 with 20 home runs and 68 RBI.
The 29-year-old Bay will be counted on to fill the power void left by Ramirez in Boston's lineup. Bay is batting .282 with 22 homers and 64 RBI.
"If it's Boston or wherever it might be, that's the silver lining, a little bit of sweet," Bay said Wednesday night. "At this time of year, I would definitely like to be part of that atmosphere."