"Nothing will happen by day's end, but we'll keep talking,'' Bay's agent, Joe Urbon, told SI.com.
The offer from the Red Sox to Bay was believed to be in the range of four years and $60 million, unidentified sources told SI.com.
Boston's window to negotiate exclusively with Bay expires at 12:01 a.m. ET Friday, when full free agency begins and players can receive and accept offers from other teams.
"It's not a surprise that a player that's gone this far wants to see what's out there,'' Red Sox GM Theo Epstein told SI.com. Epstein declined to talk to SI.com about the specifics of the Bay negotiations.
Bay, who hit .267 with 36 homers and 119 RBIs with the Red Sox last season, has said he enjoyed his year and a half in Boston and would like to return. He and the St. Louis Cardinals' Matt Holliday are the premier left fielders on the free-agent market.